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"""
This module implements some special functions that commonly appear in
combinatorial contexts (e.g. in power series); in particular,
sequences of rational numbers such as Bernoulli and Fibonacci numbers.
Factorials, binomial coefficients and related functions are located in
the separate 'factorials' module.
"""
from math import prod
from collections import defaultdict
from typing import Tuple as tTuple
from sympy.core import S, Symbol, Add, Dummy
from sympy.core.cache import cacheit
from sympy.core.expr import Expr
from sympy.core.function import ArgumentIndexError, Function, expand_mul
from sympy.core.logic import fuzzy_not
from sympy.core.mul import Mul
from sympy.core.numbers import E, I, pi, oo, Rational, Integer
from sympy.core.relational import Eq, is_le, is_gt
from sympy.external.gmpy import SYMPY_INTS
from sympy.functions.combinatorial.factorials import (binomial,
factorial, subfactorial)
from sympy.functions.elementary.exponential import log
from sympy.functions.elementary.piecewise import Piecewise
from sympy.ntheory.primetest import isprime, is_square
from sympy.polys.appellseqs import bernoulli_poly, euler_poly, genocchi_poly
from sympy.utilities.enumerative import MultisetPartitionTraverser
from sympy.utilities.exceptions import sympy_deprecation_warning
from sympy.utilities.iterables import multiset, multiset_derangements, iterable
from sympy.utilities.memoization import recurrence_memo
from sympy.utilities.misc import as_int
from mpmath import mp, workprec
from mpmath.libmp import ifib as _ifib
def _product(a, b):
return prod(range(a, b + 1))
# Dummy symbol used for computing polynomial sequences
_sym = Symbol('x')
#----------------------------------------------------------------------------#
# #
# Carmichael numbers #
# #
#----------------------------------------------------------------------------#
def _divides(p, n):
return n % p == 0
class carmichael(Function):
r"""
Carmichael Numbers:
Certain cryptographic algorithms make use of big prime numbers.
However, checking whether a big number is prime is not so easy.
Randomized prime number checking tests exist that offer a high degree of
confidence of accurate determination at low cost, such as the Fermat test.
Let 'a' be a random number between $2$ and $n - 1$, where $n$ is the
number whose primality we are testing. Then, $n$ is probably prime if it
satisfies the modular arithmetic congruence relation:
.. math :: a^{n-1} = 1 \pmod{n}
(where mod refers to the modulo operation)
If a number passes the Fermat test several times, then it is prime with a
high probability.
Unfortunately, certain composite numbers (non-primes) still pass the Fermat
test with every number smaller than themselves.
These numbers are called Carmichael numbers.
A Carmichael number will pass a Fermat primality test to every base $b$
relatively prime to the number, even though it is not actually prime.
This makes tests based on Fermat's Little Theorem less effective than
strong probable prime tests such as the Baillie-PSW primality test and
the Miller-Rabin primality test.
Examples
========
>>> from sympy import carmichael
>>> carmichael.find_first_n_carmichaels(5)
[561, 1105, 1729, 2465, 2821]
>>> carmichael.find_carmichael_numbers_in_range(0, 562)
[561]
>>> carmichael.find_carmichael_numbers_in_range(0,1000)
[561]
>>> carmichael.find_carmichael_numbers_in_range(0,2000)
[561, 1105, 1729]
References
==========
.. [1] https://en.wikipedia.org/wiki/Carmichael_number
.. [2] https://en.wikipedia.org/wiki/Fermat_primality_test
.. [3] https://www.jstor.org/stable/23248683?seq=1#metadata_info_tab_contents
"""
@staticmethod
def is_perfect_square(n):
sympy_deprecation_warning(
"""
is_perfect_square is just a wrapper around sympy.ntheory.primetest.is_square
so use that directly instead.
""",
deprecated_since_version="1.11",
active_deprecations_target='deprecated-carmichael-static-methods',
)
return is_square(n)
@staticmethod
def divides(p, n):
sympy_deprecation_warning(
"""
divides can be replaced by directly testing n % p == 0.
""",
deprecated_since_version="1.11",
active_deprecations_target='deprecated-carmichael-static-methods',
)
return n % p == 0
@staticmethod
def is_prime(n):
sympy_deprecation_warning(
"""
is_prime is just a wrapper around sympy.ntheory.primetest.isprime so use that
directly instead.
""",
deprecated_since_version="1.11",
active_deprecations_target='deprecated-carmichael-static-methods',
)
return isprime(n)
@staticmethod
def is_carmichael(n):
if n >= 0:
if (n == 1) or isprime(n) or (n % 2 == 0):
return False
divisors = [1, n]
# get divisors
divisors.extend([i for i in range(3, n // 2 + 1, 2) if n % i == 0])
for i in divisors:
if is_square(i) and i != 1:
return False
if isprime(i):
if not _divides(i - 1, n - 1):
return False
return True
else:
raise ValueError('The provided number must be greater than or equal to 0')
@staticmethod
def find_carmichael_numbers_in_range(x, y):
if 0 <= x <= y:
if x % 2 == 0:
return [i for i in range(x + 1, y, 2) if carmichael.is_carmichael(i)]
else:
return [i for i in range(x, y, 2) if carmichael.is_carmichael(i)]
else:
raise ValueError('The provided range is not valid. x and y must be non-negative integers and x <= y')
@staticmethod
def find_first_n_carmichaels(n):
i = 1
carmichaels = []
while len(carmichaels) < n:
if carmichael.is_carmichael(i):
carmichaels.append(i)
i += 2
return carmichaels
#----------------------------------------------------------------------------#
# #
# Fibonacci numbers #
# #
#----------------------------------------------------------------------------#
class fibonacci(Function):
r"""
Fibonacci numbers / Fibonacci polynomials
The Fibonacci numbers are the integer sequence defined by the
initial terms `F_0 = 0`, `F_1 = 1` and the two-term recurrence
relation `F_n = F_{n-1} + F_{n-2}`. This definition
extended to arbitrary real and complex arguments using
the formula
.. math :: F_z = \frac{\phi^z - \cos(\pi z) \phi^{-z}}{\sqrt 5}
The Fibonacci polynomials are defined by `F_1(x) = 1`,
`F_2(x) = x`, and `F_n(x) = x*F_{n-1}(x) + F_{n-2}(x)` for `n > 2`.
For all positive integers `n`, `F_n(1) = F_n`.
* ``fibonacci(n)`` gives the `n^{th}` Fibonacci number, `F_n`
* ``fibonacci(n, x)`` gives the `n^{th}` Fibonacci polynomial in `x`, `F_n(x)`
Examples
========
>>> from sympy import fibonacci, Symbol
>>> [fibonacci(x) for x in range(11)]
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
>>> fibonacci(5, Symbol('t'))
t**4 + 3*t**2 + 1
See Also
========
bell, bernoulli, catalan, euler, harmonic, lucas, genocchi, partition, tribonacci
References
==========
.. [1] https://en.wikipedia.org/wiki/Fibonacci_number
.. [2] https://mathworld.wolfram.com/FibonacciNumber.html
"""
@staticmethod
def _fib(n):
return _ifib(n)
@staticmethod
@recurrence_memo([None, S.One, _sym])
def _fibpoly(n, prev):
return (prev[-2] + _sym*prev[-1]).expand()
@classmethod
def eval(cls, n, sym=None):
if n is S.Infinity:
return S.Infinity
if n.is_Integer:
if sym is None:
n = int(n)
if n < 0:
return S.NegativeOne**(n + 1) * fibonacci(-n)
else:
return Integer(cls._fib(n))
else:
if n < 1:
raise ValueError("Fibonacci polynomials are defined "
"only for positive integer indices.")
return cls._fibpoly(n).subs(_sym, sym)
def _eval_rewrite_as_sqrt(self, n, **kwargs):
from sympy.functions.elementary.miscellaneous import sqrt
return 2**(-n)*sqrt(5)*((1 + sqrt(5))**n - (-sqrt(5) + 1)**n) / 5
def _eval_rewrite_as_GoldenRatio(self,n, **kwargs):
return (S.GoldenRatio**n - 1/(-S.GoldenRatio)**n)/(2*S.GoldenRatio-1)
#----------------------------------------------------------------------------#
# #
# Lucas numbers #
# #
#----------------------------------------------------------------------------#
class lucas(Function):
"""
Lucas numbers
Lucas numbers satisfy a recurrence relation similar to that of
the Fibonacci sequence, in which each term is the sum of the
preceding two. They are generated by choosing the initial
values `L_0 = 2` and `L_1 = 1`.
* ``lucas(n)`` gives the `n^{th}` Lucas number
Examples
========
>>> from sympy import lucas
>>> [lucas(x) for x in range(11)]
[2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123]
See Also
========
bell, bernoulli, catalan, euler, fibonacci, harmonic, genocchi, partition, tribonacci
References
==========
.. [1] https://en.wikipedia.org/wiki/Lucas_number
.. [2] https://mathworld.wolfram.com/LucasNumber.html
"""
@classmethod
def eval(cls, n):
if n is S.Infinity:
return S.Infinity
if n.is_Integer:
return fibonacci(n + 1) + fibonacci(n - 1)
def _eval_rewrite_as_sqrt(self, n, **kwargs):
from sympy.functions.elementary.miscellaneous import sqrt
return 2**(-n)*((1 + sqrt(5))**n + (-sqrt(5) + 1)**n)
#----------------------------------------------------------------------------#
# #
# Tribonacci numbers #
# #
#----------------------------------------------------------------------------#
class tribonacci(Function):
r"""
Tribonacci numbers / Tribonacci polynomials
The Tribonacci numbers are the integer sequence defined by the
initial terms `T_0 = 0`, `T_1 = 1`, `T_2 = 1` and the three-term
recurrence relation `T_n = T_{n-1} + T_{n-2} + T_{n-3}`.
The Tribonacci polynomials are defined by `T_0(x) = 0`, `T_1(x) = 1`,
`T_2(x) = x^2`, and `T_n(x) = x^2 T_{n-1}(x) + x T_{n-2}(x) + T_{n-3}(x)`
for `n > 2`. For all positive integers `n`, `T_n(1) = T_n`.
* ``tribonacci(n)`` gives the `n^{th}` Tribonacci number, `T_n`
* ``tribonacci(n, x)`` gives the `n^{th}` Tribonacci polynomial in `x`, `T_n(x)`
Examples
========
>>> from sympy import tribonacci, Symbol
>>> [tribonacci(x) for x in range(11)]
[0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149]
>>> tribonacci(5, Symbol('t'))
t**8 + 3*t**5 + 3*t**2
See Also
========
bell, bernoulli, catalan, euler, fibonacci, harmonic, lucas, genocchi, partition
References
==========
.. [1] https://en.wikipedia.org/wiki/Generalizations_of_Fibonacci_numbers#Tribonacci_numbers
.. [2] https://mathworld.wolfram.com/TribonacciNumber.html
.. [3] https://oeis.org/A000073
"""
@staticmethod
@recurrence_memo([S.Zero, S.One, S.One])
def _trib(n, prev):
return (prev[-3] + prev[-2] + prev[-1])
@staticmethod
@recurrence_memo([S.Zero, S.One, _sym**2])
def _tribpoly(n, prev):
return (prev[-3] + _sym*prev[-2] + _sym**2*prev[-1]).expand()
@classmethod
def eval(cls, n, sym=None):
if n is S.Infinity:
return S.Infinity
if n.is_Integer:
n = int(n)
if n < 0:
raise ValueError("Tribonacci polynomials are defined "
"only for non-negative integer indices.")
if sym is None:
return Integer(cls._trib(n))
else:
return cls._tribpoly(n).subs(_sym, sym)
def _eval_rewrite_as_sqrt(self, n, **kwargs):
from sympy.functions.elementary.miscellaneous import cbrt, sqrt
w = (-1 + S.ImaginaryUnit * sqrt(3)) / 2
a = (1 + cbrt(19 + 3*sqrt(33)) + cbrt(19 - 3*sqrt(33))) / 3
b = (1 + w*cbrt(19 + 3*sqrt(33)) + w**2*cbrt(19 - 3*sqrt(33))) / 3
c = (1 + w**2*cbrt(19 + 3*sqrt(33)) + w*cbrt(19 - 3*sqrt(33))) / 3
Tn = (a**(n + 1)/((a - b)*(a - c))
+ b**(n + 1)/((b - a)*(b - c))
+ c**(n + 1)/((c - a)*(c - b)))
return Tn
def _eval_rewrite_as_TribonacciConstant(self, n, **kwargs):
from sympy.functions.elementary.integers import floor
from sympy.functions.elementary.miscellaneous import cbrt, sqrt
b = cbrt(586 + 102*sqrt(33))
Tn = 3 * b * S.TribonacciConstant**n / (b**2 - 2*b + 4)
return floor(Tn + S.Half)
#----------------------------------------------------------------------------#
# #
# Bernoulli numbers #
# #
#----------------------------------------------------------------------------#
class bernoulli(Function):
r"""
Bernoulli numbers / Bernoulli polynomials / Bernoulli function
The Bernoulli numbers are a sequence of rational numbers
defined by `B_0 = 1` and the recursive relation (`n > 0`):
.. math :: n+1 = \sum_{k=0}^n \binom{n+1}{k} B_k
They are also commonly defined by their exponential generating
function, which is `\frac{x}{1 - e^{-x}}`. For odd indices > 1,
the Bernoulli numbers are zero.
The Bernoulli polynomials satisfy the analogous formula:
.. math :: B_n(x) = \sum_{k=0}^n (-1)^k \binom{n}{k} B_k x^{n-k}
Bernoulli numbers and Bernoulli polynomials are related as
`B_n(1) = B_n`.
The generalized Bernoulli function `\operatorname{B}(s, a)`
is defined for any complex `s` and `a`, except where `a` is a
nonpositive integer and `s` is not a nonnegative integer. It is
an entire function of `s` for fixed `a`, related to the Hurwitz
zeta function by
.. math:: \operatorname{B}(s, a) = \begin{cases}
-s \zeta(1-s, a) & s \ne 0 \\ 1 & s = 0 \end{cases}
When `s` is a nonnegative integer this function reduces to the
Bernoulli polynomials: `\operatorname{B}(n, x) = B_n(x)`. When
`a` is omitted it is assumed to be 1, yielding the (ordinary)
Bernoulli function which interpolates the Bernoulli numbers and is
related to the Riemann zeta function.
We compute Bernoulli numbers using Ramanujan's formula:
.. math :: B_n = \frac{A(n) - S(n)}{\binom{n+3}{n}}
where:
.. math :: A(n) = \begin{cases} \frac{n+3}{3} &
n \equiv 0\ \text{or}\ 2 \pmod{6} \\
-\frac{n+3}{6} & n \equiv 4 \pmod{6} \end{cases}
and:
.. math :: S(n) = \sum_{k=1}^{[n/6]} \binom{n+3}{n-6k} B_{n-6k}
This formula is similar to the sum given in the definition, but
cuts `\frac{2}{3}` of the terms. For Bernoulli polynomials, we use
Appell sequences.
For `n` a nonnegative integer and `s`, `a`, `x` arbitrary complex numbers,
* ``bernoulli(n)`` gives the nth Bernoulli number, `B_n`
* ``bernoulli(s)`` gives the Bernoulli function `\operatorname{B}(s)`
* ``bernoulli(n, x)`` gives the nth Bernoulli polynomial in `x`, `B_n(x)`
* ``bernoulli(s, a)`` gives the generalized Bernoulli function
`\operatorname{B}(s, a)`
.. versionchanged:: 1.12
``bernoulli(1)`` gives `+\frac{1}{2}` instead of `-\frac{1}{2}`.
This choice of value confers several theoretical advantages [5]_,
including the extension to complex parameters described above
which this function now implements. The previous behavior, defined
only for nonnegative integers `n`, can be obtained with
``(-1)**n*bernoulli(n)``.
Examples
========
>>> from sympy import bernoulli
>>> from sympy.abc import x
>>> [bernoulli(n) for n in range(11)]
[1, 1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66]
>>> bernoulli(1000001)
0
>>> bernoulli(3, x)
x**3 - 3*x**2/2 + x/2
See Also
========
andre, bell, catalan, euler, fibonacci, harmonic, lucas, genocchi,
partition, tribonacci, sympy.polys.appellseqs.bernoulli_poly
References
==========
.. [1] https://en.wikipedia.org/wiki/Bernoulli_number
.. [2] https://en.wikipedia.org/wiki/Bernoulli_polynomial
.. [3] https://mathworld.wolfram.com/BernoulliNumber.html
.. [4] https://mathworld.wolfram.com/BernoulliPolynomial.html
.. [5] Peter Luschny, "The Bernoulli Manifesto",
https://luschny.de/math/zeta/The-Bernoulli-Manifesto.html
.. [6] Peter Luschny, "An introduction to the Bernoulli function",
https://arxiv.org/abs/2009.06743
"""
args: tTuple[Integer]
# Calculates B_n for positive even n
@staticmethod
def _calc_bernoulli(n):
s = 0
a = int(binomial(n + 3, n - 6))
for j in range(1, n//6 + 1):
s += a * bernoulli(n - 6*j)
# Avoid computing each binomial coefficient from scratch
a *= _product(n - 6 - 6*j + 1, n - 6*j)
a //= _product(6*j + 4, 6*j + 9)
if n % 6 == 4:
s = -Rational(n + 3, 6) - s
else:
s = Rational(n + 3, 3) - s
return s / binomial(n + 3, n)
# We implement a specialized memoization scheme to handle each
# case modulo 6 separately
_cache = {0: S.One, 2: Rational(1, 6), 4: Rational(-1, 30)}
_highest = {0: 0, 2: 2, 4: 4}
@classmethod
def eval(cls, n, x=None):
if x is S.One:
return cls(n)
elif n.is_zero:
return S.One
elif n.is_integer is False or n.is_nonnegative is False:
if x is not None and x.is_Integer and x.is_nonpositive:
return S.NaN
return
# Bernoulli numbers
elif x is None:
if n is S.One:
return S.Half
elif n.is_odd and (n-1).is_positive:
return S.Zero
elif n.is_Number:
n = int(n)
# Use mpmath for enormous Bernoulli numbers
if n > 500:
p, q = mp.bernfrac(n)
return Rational(int(p), int(q))
case = n % 6
highest_cached = cls._highest[case]
if n <= highest_cached:
return cls._cache[n]
# To avoid excessive recursion when, say, bernoulli(1000) is
# requested, calculate and cache the entire sequence ... B_988,
# B_994, B_1000 in increasing order
for i in range(highest_cached + 6, n + 6, 6):
b = cls._calc_bernoulli(i)
cls._cache[i] = b
cls._highest[case] = i
return b
# Bernoulli polynomials
elif n.is_Number:
return bernoulli_poly(n, x)
def _eval_rewrite_as_zeta(self, n, x=1, **kwargs):
from sympy.functions.special.zeta_functions import zeta
return Piecewise((1, Eq(n, 0)), (-n * zeta(1-n, x), True))
def _eval_evalf(self, prec):
if not all(x.is_number for x in self.args):
return
n = self.args[0]._to_mpmath(prec)
x = (self.args[1] if len(self.args) > 1 else S.One)._to_mpmath(prec)
with workprec(prec):
if n == 0:
res = mp.mpf(1)
elif n == 1:
res = x - mp.mpf(0.5)
elif mp.isint(n) and n >= 0:
res = mp.bernoulli(n) if x == 1 else mp.bernpoly(n, x)
else:
res = -n * mp.zeta(1-n, x)
return Expr._from_mpmath(res, prec)
#----------------------------------------------------------------------------#
# #
# Bell numbers #
# #
#----------------------------------------------------------------------------#
class bell(Function):
r"""
Bell numbers / Bell polynomials
The Bell numbers satisfy `B_0 = 1` and
.. math:: B_n = \sum_{k=0}^{n-1} \binom{n-1}{k} B_k.
They are also given by:
.. math:: B_n = \frac{1}{e} \sum_{k=0}^{\infty} \frac{k^n}{k!}.
The Bell polynomials are given by `B_0(x) = 1` and
.. math:: B_n(x) = x \sum_{k=1}^{n-1} \binom{n-1}{k-1} B_{k-1}(x).
The second kind of Bell polynomials (are sometimes called "partial" Bell
polynomials or incomplete Bell polynomials) are defined as
.. math:: B_{n,k}(x_1, x_2,\dotsc x_{n-k+1}) =
\sum_{j_1+j_2+j_2+\dotsb=k \atop j_1+2j_2+3j_2+\dotsb=n}
\frac{n!}{j_1!j_2!\dotsb j_{n-k+1}!}
\left(\frac{x_1}{1!} \right)^{j_1}
\left(\frac{x_2}{2!} \right)^{j_2} \dotsb
\left(\frac{x_{n-k+1}}{(n-k+1)!} \right) ^{j_{n-k+1}}.
* ``bell(n)`` gives the `n^{th}` Bell number, `B_n`.
* ``bell(n, x)`` gives the `n^{th}` Bell polynomial, `B_n(x)`.
* ``bell(n, k, (x1, x2, ...))`` gives Bell polynomials of the second kind,
`B_{n,k}(x_1, x_2, \dotsc, x_{n-k+1})`.
Notes
=====
Not to be confused with Bernoulli numbers and Bernoulli polynomials,
which use the same notation.
Examples
========
>>> from sympy import bell, Symbol, symbols
>>> [bell(n) for n in range(11)]
[1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975]
>>> bell(30)
846749014511809332450147
>>> bell(4, Symbol('t'))
t**4 + 6*t**3 + 7*t**2 + t
>>> bell(6, 2, symbols('x:6')[1:])
6*x1*x5 + 15*x2*x4 + 10*x3**2
See Also
========
bernoulli, catalan, euler, fibonacci, harmonic, lucas, genocchi, partition, tribonacci
References
==========
.. [1] https://en.wikipedia.org/wiki/Bell_number
.. [2] https://mathworld.wolfram.com/BellNumber.html
.. [3] https://mathworld.wolfram.com/BellPolynomial.html
"""
@staticmethod
@recurrence_memo([1, 1])
def _bell(n, prev):
s = 1
a = 1
for k in range(1, n):
a = a * (n - k) // k
s += a * prev[k]
return s
@staticmethod
@recurrence_memo([S.One, _sym])
def _bell_poly(n, prev):
s = 1
a = 1
for k in range(2, n + 1):
a = a * (n - k + 1) // (k - 1)
s += a * prev[k - 1]
return expand_mul(_sym * s)
@staticmethod
def _bell_incomplete_poly(n, k, symbols):
r"""
The second kind of Bell polynomials (incomplete Bell polynomials).
Calculated by recurrence formula:
.. math:: B_{n,k}(x_1, x_2, \dotsc, x_{n-k+1}) =
\sum_{m=1}^{n-k+1}
\x_m \binom{n-1}{m-1} B_{n-m,k-1}(x_1, x_2, \dotsc, x_{n-m-k})
where
`B_{0,0} = 1;`
`B_{n,0} = 0; for n \ge 1`
`B_{0,k} = 0; for k \ge 1`
"""
if (n == 0) and (k == 0):
return S.One
elif (n == 0) or (k == 0):
return S.Zero
s = S.Zero
a = S.One
for m in range(1, n - k + 2):
s += a * bell._bell_incomplete_poly(
n - m, k - 1, symbols) * symbols[m - 1]
a = a * (n - m) / m
return expand_mul(s)
@classmethod
def eval(cls, n, k_sym=None, symbols=None):
if n is S.Infinity:
if k_sym is None:
return S.Infinity
else:
raise ValueError("Bell polynomial is not defined")
if n.is_negative or n.is_integer is False:
raise ValueError("a non-negative integer expected")
if n.is_Integer and n.is_nonnegative:
if k_sym is None:
return Integer(cls._bell(int(n)))
elif symbols is None:
return cls._bell_poly(int(n)).subs(_sym, k_sym)
else:
r = cls._bell_incomplete_poly(int(n), int(k_sym), symbols)
return r
def _eval_rewrite_as_Sum(self, n, k_sym=None, symbols=None, **kwargs):
from sympy.concrete.summations import Sum
if (k_sym is not None) or (symbols is not None):
return self
# Dobinski's formula
if not n.is_nonnegative:
return self
k = Dummy('k', integer=True, nonnegative=True)
return 1 / E * Sum(k**n / factorial(k), (k, 0, S.Infinity))
#----------------------------------------------------------------------------#
# #
# Harmonic numbers #
# #
#----------------------------------------------------------------------------#
class harmonic(Function):
r"""
Harmonic numbers
The nth harmonic number is given by `\operatorname{H}_{n} =
1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n}`.
More generally:
.. math:: \operatorname{H}_{n,m} = \sum_{k=1}^{n} \frac{1}{k^m}
As `n \rightarrow \infty`, `\operatorname{H}_{n,m} \rightarrow \zeta(m)`,
the Riemann zeta function.
* ``harmonic(n)`` gives the nth harmonic number, `\operatorname{H}_n`
* ``harmonic(n, m)`` gives the nth generalized harmonic number
of order `m`, `\operatorname{H}_{n,m}`, where
``harmonic(n) == harmonic(n, 1)``
This function can be extended to complex `n` and `m` where `n` is not a
negative integer or `m` is a nonpositive integer as
.. math:: \operatorname{H}_{n,m} = \begin{cases} \zeta(m) - \zeta(m, n+1)
& m \ne 1 \\ \psi(n+1) + \gamma & m = 1 \end{cases}
Examples
========
>>> from sympy import harmonic, oo
>>> [harmonic(n) for n in range(6)]
[0, 1, 3/2, 11/6, 25/12, 137/60]
>>> [harmonic(n, 2) for n in range(6)]
[0, 1, 5/4, 49/36, 205/144, 5269/3600]
>>> harmonic(oo, 2)
pi**2/6
>>> from sympy import Symbol, Sum
>>> n = Symbol("n")
>>> harmonic(n).rewrite(Sum)
Sum(1/_k, (_k, 1, n))
We can evaluate harmonic numbers for all integral and positive
rational arguments:
>>> from sympy import S, expand_func, simplify
>>> harmonic(8)
761/280
>>> harmonic(11)
83711/27720
>>> H = harmonic(1/S(3))
>>> H
harmonic(1/3)
>>> He = expand_func(H)
>>> He
-log(6) - sqrt(3)*pi/6 + 2*Sum(log(sin(_k*pi/3))*cos(2*_k*pi/3), (_k, 1, 1))
+ 3*Sum(1/(3*_k + 1), (_k, 0, 0))
>>> He.doit()
-log(6) - sqrt(3)*pi/6 - log(sqrt(3)/2) + 3
>>> H = harmonic(25/S(7))
>>> He = simplify(expand_func(H).doit())
>>> He
log(sin(2*pi/7)**(2*cos(16*pi/7))/(14*sin(pi/7)**(2*cos(pi/7))*cos(pi/14)**(2*sin(pi/14)))) + pi*tan(pi/14)/2 + 30247/9900
>>> He.n(40)
1.983697455232980674869851942390639915940
>>> harmonic(25/S(7)).n(40)
1.983697455232980674869851942390639915940
We can rewrite harmonic numbers in terms of polygamma functions:
>>> from sympy import digamma, polygamma
>>> m = Symbol("m", integer=True, positive=True)
>>> harmonic(n).rewrite(digamma)
polygamma(0, n + 1) + EulerGamma
>>> harmonic(n).rewrite(polygamma)
polygamma(0, n + 1) + EulerGamma
>>> harmonic(n,3).rewrite(polygamma)
polygamma(2, n + 1)/2 + zeta(3)
>>> simplify(harmonic(n,m).rewrite(polygamma))
Piecewise((polygamma(0, n + 1) + EulerGamma, Eq(m, 1)),
(-(-1)**m*polygamma(m - 1, n + 1)/factorial(m - 1) + zeta(m), True))
Integer offsets in the argument can be pulled out:
>>> from sympy import expand_func
>>> expand_func(harmonic(n+4))
harmonic(n) + 1/(n + 4) + 1/(n + 3) + 1/(n + 2) + 1/(n + 1)
>>> expand_func(harmonic(n-4))
harmonic(n) - 1/(n - 1) - 1/(n - 2) - 1/(n - 3) - 1/n
Some limits can be computed as well:
>>> from sympy import limit, oo
>>> limit(harmonic(n), n, oo)
oo
>>> limit(harmonic(n, 2), n, oo)
pi**2/6
>>> limit(harmonic(n, 3), n, oo)
zeta(3)
For `m > 1`, `H_{n,m}` tends to `\zeta(m)` in the limit of infinite `n`:
>>> m = Symbol("m", positive=True)
>>> limit(harmonic(n, m+1), n, oo)
zeta(m + 1)
See Also
========
bell, bernoulli, catalan, euler, fibonacci, lucas, genocchi, partition, tribonacci
References
==========
.. [1] https://en.wikipedia.org/wiki/Harmonic_number
.. [2] https://functions.wolfram.com/GammaBetaErf/HarmonicNumber/
.. [3] https://functions.wolfram.com/GammaBetaErf/HarmonicNumber2/
"""
@classmethod
def eval(cls, n, m=None):
from sympy.functions.special.zeta_functions import zeta
if m is S.One:
return cls(n)
if m is None:
m = S.One
if n.is_zero:
return S.Zero
elif m.is_zero:
return n
elif n is S.Infinity:
if m.is_negative:
return S.NaN
elif is_le(m, S.One):
return S.Infinity
elif is_gt(m, S.One):
return zeta(m)
elif m.is_Integer and m.is_nonpositive:
return (bernoulli(1-m, n+1) - bernoulli(1-m)) / (1-m)
elif n.is_Integer:
if n.is_negative and (m.is_integer is False or m.is_nonpositive is False):
return S.ComplexInfinity if m is S.One else S.NaN
if n.is_nonnegative:
return Add(*(k**(-m) for k in range(1, int(n)+1)))
def _eval_rewrite_as_polygamma(self, n, m=S.One, **kwargs):
from sympy.functions.special.gamma_functions import gamma, polygamma
if m.is_integer and m.is_positive:
return Piecewise((polygamma(0, n+1) + S.EulerGamma, Eq(m, 1)),
(S.NegativeOne**m * (polygamma(m-1, 1) - polygamma(m-1, n+1)) /
gamma(m), True))
def _eval_rewrite_as_digamma(self, n, m=1, **kwargs):
from sympy.functions.special.gamma_functions import polygamma
return self.rewrite(polygamma)
def _eval_rewrite_as_trigamma(self, n, m=1, **kwargs):
from sympy.functions.special.gamma_functions import polygamma
return self.rewrite(polygamma)
def _eval_rewrite_as_Sum(self, n, m=None, **kwargs):
from sympy.concrete.summations import Sum
k = Dummy("k", integer=True)
if m is None:
m = S.One
return Sum(k**(-m), (k, 1, n))
def _eval_rewrite_as_zeta(self, n, m=S.One, **kwargs):
from sympy.functions.special.zeta_functions import zeta
from sympy.functions.special.gamma_functions import digamma
return Piecewise((digamma(n + 1) + S.EulerGamma, Eq(m, 1)),
(zeta(m) - zeta(m, n+1), True))
def _eval_expand_func(self, **hints):
from sympy.concrete.summations import Sum
n = self.args[0]
m = self.args[1] if len(self.args) == 2 else 1
if m == S.One:
if n.is_Add:
off = n.args[0]
nnew = n - off
if off.is_Integer and off.is_positive:
result = [S.One/(nnew + i) for i in range(off, 0, -1)] + [harmonic(nnew)]
return Add(*result)
elif off.is_Integer and off.is_negative:
result = [-S.One/(nnew + i) for i in range(0, off, -1)] + [harmonic(nnew)]
return Add(*result)
if n.is_Rational:
# Expansions for harmonic numbers at general rational arguments (u + p/q)
# Split n as u + p/q with p < q
p, q = n.as_numer_denom()
u = p // q
p = p - u * q
if u.is_nonnegative and p.is_positive and q.is_positive and p < q:
from sympy.functions.elementary.exponential import log
from sympy.functions.elementary.integers import floor
from sympy.functions.elementary.trigonometric import sin, cos, cot
k = Dummy("k")
t1 = q * Sum(1 / (q * k + p), (k, 0, u))
t2 = 2 * Sum(cos((2 * pi * p * k) / S(q)) *
log(sin((pi * k) / S(q))),
(k, 1, floor((q - 1) / S(2))))
t3 = (pi / 2) * cot((pi * p) / q) + log(2 * q)
return t1 + t2 - t3
return self
def _eval_rewrite_as_tractable(self, n, m=1, limitvar=None, **kwargs):
from sympy.functions.special.zeta_functions import zeta
from sympy.functions.special.gamma_functions import polygamma
pg = self.rewrite(polygamma)
if not isinstance(pg, harmonic):
return pg.rewrite("tractable", deep=True)
arg = m - S.One
if arg.is_nonzero:
return (zeta(m) - zeta(m, n+1)).rewrite("tractable", deep=True)
def _eval_evalf(self, prec):
if not all(x.is_number for x in self.args):
return
n = self.args[0]._to_mpmath(prec)
m = (self.args[1] if len(self.args) > 1 else S.One)._to_mpmath(prec)
if mp.isint(n) and n < 0:
return S.NaN
with workprec(prec):
if m == 1:
res = mp.harmonic(n)
else:
res = mp.zeta(m) - mp.zeta(m, n+1)
return Expr._from_mpmath(res, prec)
def fdiff(self, argindex=1):
from sympy.functions.special.zeta_functions import zeta
if len(self.args) == 2:
n, m = self.args
else:
n, m = self.args + (1,)
if argindex == 1:
return m * zeta(m+1, n+1)
else:
raise ArgumentIndexError
#----------------------------------------------------------------------------#
# #
# Euler numbers #
# #
#----------------------------------------------------------------------------#
class euler(Function):
r"""
Euler numbers / Euler polynomials / Euler function
The Euler numbers are given by:
.. math:: E_{2n} = I \sum_{k=1}^{2n+1} \sum_{j=0}^k \binom{k}{j}
\frac{(-1)^j (k-2j)^{2n+1}}{2^k I^k k}
.. math:: E_{2n+1} = 0
Euler numbers and Euler polynomials are related by
.. math:: E_n = 2^n E_n\left(\frac{1}{2}\right).
We compute symbolic Euler polynomials using Appell sequences,
but numerical evaluation of the Euler polynomial is computed
more efficiently (and more accurately) using the mpmath library.
The Euler polynomials are special cases of the generalized Euler function,
related to the Genocchi function as
.. math:: \operatorname{E}(s, a) = -\frac{\operatorname{G}(s+1, a)}{s+1}
with the limit of `\psi\left(\frac{a+1}{2}\right) - \psi\left(\frac{a}{2}\right)`
being taken when `s = -1`. The (ordinary) Euler function interpolating
the Euler numbers is then obtained as
`\operatorname{E}(s) = 2^s \operatorname{E}\left(s, \frac{1}{2}\right)`.
* ``euler(n)`` gives the nth Euler number `E_n`.
* ``euler(s)`` gives the Euler function `\operatorname{E}(s)`.
* ``euler(n, x)`` gives the nth Euler polynomial `E_n(x)`.
* ``euler(s, a)`` gives the generalized Euler function `\operatorname{E}(s, a)`.
Examples
========
>>> from sympy import euler, Symbol, S
>>> [euler(n) for n in range(10)]
[1, 0, -1, 0, 5, 0, -61, 0, 1385, 0]
>>> [2**n*euler(n,1) for n in range(10)]
[1, 1, 0, -2, 0, 16, 0, -272, 0, 7936]
>>> n = Symbol("n")
>>> euler(n + 2*n)
euler(3*n)
>>> x = Symbol("x")
>>> euler(n, x)
euler(n, x)
>>> euler(0, x)
1
>>> euler(1, x)
x - 1/2
>>> euler(2, x)
x**2 - x
>>> euler(3, x)
x**3 - 3*x**2/2 + 1/4
>>> euler(4, x)
x**4 - 2*x**3 + x
>>> euler(12, S.Half)
2702765/4096
>>> euler(12)
2702765
See Also
========
andre, bell, bernoulli, catalan, fibonacci, harmonic, lucas, genocchi,
partition, tribonacci, sympy.polys.appellseqs.euler_poly
References
==========
.. [1] https://en.wikipedia.org/wiki/Euler_numbers
.. [2] https://mathworld.wolfram.com/EulerNumber.html
.. [3] https://en.wikipedia.org/wiki/Alternating_permutation
.. [4] https://mathworld.wolfram.com/AlternatingPermutation.html
"""
@classmethod
def eval(cls, n, x=None):
if n.is_zero:
return S.One
elif n is S.NegativeOne:
if x is None:
return S.Pi/2
from sympy.functions.special.gamma_functions import digamma
return digamma((x+1)/2) - digamma(x/2)
elif n.is_integer is False or n.is_nonnegative is False:
return
# Euler numbers
elif x is None:
if n.is_odd and n.is_positive:
return S.Zero
elif n.is_Number:
from mpmath import mp
n = n._to_mpmath(mp.prec)
res = mp.eulernum(n, exact=True)
return Integer(res)
# Euler polynomials
elif n.is_Number:
return euler_poly(n, x)
def _eval_rewrite_as_Sum(self, n, x=None, **kwargs):
from sympy.concrete.summations import Sum
if x is None and n.is_even:
k = Dummy("k", integer=True)
j = Dummy("j", integer=True)
n = n / 2
Em = (S.ImaginaryUnit * Sum(Sum(binomial(k, j) * (S.NegativeOne**j *
(k - 2*j)**(2*n + 1)) /
(2**k*S.ImaginaryUnit**k * k), (j, 0, k)), (k, 1, 2*n + 1)))
return Em
if x:
k = Dummy("k", integer=True)
return Sum(binomial(n, k)*euler(k)/2**k*(x - S.Half)**(n - k), (k, 0, n))
def _eval_rewrite_as_genocchi(self, n, x=None, **kwargs):
if x is None:
return Piecewise((S.Pi/2, Eq(n, -1)),
(-2**n * genocchi(n+1, S.Half) / (n+1), True))
from sympy.functions.special.gamma_functions import digamma
return Piecewise((digamma((x+1)/2) - digamma(x/2), Eq(n, -1)),
(-genocchi(n+1, x) / (n+1), True))
def _eval_evalf(self, prec):
if not all(i.is_number for i in self.args):
return
from mpmath import mp
m, x = (self.args[0], None) if len(self.args) == 1 else self.args
m = m._to_mpmath(prec)
if x is not None:
x = x._to_mpmath(prec)
with workprec(prec):
if mp.isint(m) and m >= 0:
res = mp.eulernum(m) if x is None else mp.eulerpoly(m, x)
else:
if m == -1:
res = mp.pi if x is None else mp.digamma((x+1)/2) - mp.digamma(x/2)
else:
y = 0.5 if x is None else x
res = 2 * (mp.zeta(-m, y) - 2**(m+1) * mp.zeta(-m, (y+1)/2))
if x is None:
res *= 2**m
return Expr._from_mpmath(res, prec)
#----------------------------------------------------------------------------#
# #
# Catalan numbers #
# #
#----------------------------------------------------------------------------#
class catalan(Function):
r"""
Catalan numbers
The `n^{th}` catalan number is given by:
.. math :: C_n = \frac{1}{n+1} \binom{2n}{n}
* ``catalan(n)`` gives the `n^{th}` Catalan number, `C_n`
Examples
========
>>> from sympy import (Symbol, binomial, gamma, hyper,
... catalan, diff, combsimp, Rational, I)
>>> [catalan(i) for i in range(1,10)]
[1, 2, 5, 14, 42, 132, 429, 1430, 4862]
>>> n = Symbol("n", integer=True)
>>> catalan(n)
catalan(n)
Catalan numbers can be transformed into several other, identical
expressions involving other mathematical functions
>>> catalan(n).rewrite(binomial)
binomial(2*n, n)/(n + 1)
>>> catalan(n).rewrite(gamma)
4**n*gamma(n + 1/2)/(sqrt(pi)*gamma(n + 2))
>>> catalan(n).rewrite(hyper)
hyper((1 - n, -n), (2,), 1)
For some non-integer values of n we can get closed form
expressions by rewriting in terms of gamma functions:
>>> catalan(Rational(1, 2)).rewrite(gamma)
8/(3*pi)
We can differentiate the Catalan numbers C(n) interpreted as a
continuous real function in n:
>>> diff(catalan(n), n)
(polygamma(0, n + 1/2) - polygamma(0, n + 2) + log(4))*catalan(n)
As a more advanced example consider the following ratio
between consecutive numbers:
>>> combsimp((catalan(n + 1)/catalan(n)).rewrite(binomial))
2*(2*n + 1)/(n + 2)
The Catalan numbers can be generalized to complex numbers:
>>> catalan(I).rewrite(gamma)
4**I*gamma(1/2 + I)/(sqrt(pi)*gamma(2 + I))
and evaluated with arbitrary precision:
>>> catalan(I).evalf(20)
0.39764993382373624267 - 0.020884341620842555705*I
See Also
========
andre, bell, bernoulli, euler, fibonacci, harmonic, lucas, genocchi,
partition, tribonacci, sympy.functions.combinatorial.factorials.binomial
References
==========
.. [1] https://en.wikipedia.org/wiki/Catalan_number
.. [2] https://mathworld.wolfram.com/CatalanNumber.html
.. [3] https://functions.wolfram.com/GammaBetaErf/CatalanNumber/
.. [4] http://geometer.org/mathcircles/catalan.pdf
"""
@classmethod
def eval(cls, n):
from sympy.functions.special.gamma_functions import gamma
if (n.is_Integer and n.is_nonnegative) or \
(n.is_noninteger and n.is_negative):
return 4**n*gamma(n + S.Half)/(gamma(S.Half)*gamma(n + 2))
if (n.is_integer and n.is_negative):
if (n + 1).is_negative:
return S.Zero
if (n + 1).is_zero:
return Rational(-1, 2)
def fdiff(self, argindex=1):
from sympy.functions.elementary.exponential import log
from sympy.functions.special.gamma_functions import polygamma
n = self.args[0]
return catalan(n)*(polygamma(0, n + S.Half) - polygamma(0, n + 2) + log(4))
def _eval_rewrite_as_binomial(self, n, **kwargs):
return binomial(2*n, n)/(n + 1)
def _eval_rewrite_as_factorial(self, n, **kwargs):
return factorial(2*n) / (factorial(n+1) * factorial(n))
def _eval_rewrite_as_gamma(self, n, piecewise=True, **kwargs):
from sympy.functions.special.gamma_functions import gamma
# The gamma function allows to generalize Catalan numbers to complex n
return 4**n*gamma(n + S.Half)/(gamma(S.Half)*gamma(n + 2))
def _eval_rewrite_as_hyper(self, n, **kwargs):
from sympy.functions.special.hyper import hyper
return hyper([1 - n, -n], [2], 1)
def _eval_rewrite_as_Product(self, n, **kwargs):
from sympy.concrete.products import Product
if not (n.is_integer and n.is_nonnegative):
return self
k = Dummy('k', integer=True, positive=True)
return Product((n + k) / k, (k, 2, n))
def _eval_is_integer(self):
if self.args[0].is_integer and self.args[0].is_nonnegative:
return True
def _eval_is_positive(self):
if self.args[0].is_nonnegative:
return True
def _eval_is_composite(self):
if self.args[0].is_integer and (self.args[0] - 3).is_positive:
return True
def _eval_evalf(self, prec):
from sympy.functions.special.gamma_functions import gamma
if self.args[0].is_number:
return self.rewrite(gamma)._eval_evalf(prec)
#----------------------------------------------------------------------------#
# #
# Genocchi numbers #
# #
#----------------------------------------------------------------------------#
class genocchi(Function):
r"""
Genocchi numbers / Genocchi polynomials / Genocchi function
The Genocchi numbers are a sequence of integers `G_n` that satisfy the
relation:
.. math:: \frac{-2t}{1 + e^{-t}} = \sum_{n=0}^\infty \frac{G_n t^n}{n!}
They are related to the Bernoulli numbers by
.. math:: G_n = 2 (1 - 2^n) B_n
and generalize like the Bernoulli numbers to the Genocchi polynomials and
function as
.. math:: \operatorname{G}(s, a) = 2 \left(\operatorname{B}(s, a) -
2^s \operatorname{B}\left(s, \frac{a+1}{2}\right)\right)
.. versionchanged:: 1.12
``genocchi(1)`` gives `-1` instead of `1`.
Examples
========
>>> from sympy import genocchi, Symbol
>>> [genocchi(n) for n in range(9)]
[0, -1, -1, 0, 1, 0, -3, 0, 17]
>>> n = Symbol('n', integer=True, positive=True)
>>> genocchi(2*n + 1)
0
>>> x = Symbol('x')
>>> genocchi(4, x)
-4*x**3 + 6*x**2 - 1
See Also
========
bell, bernoulli, catalan, euler, fibonacci, harmonic, lucas, partition, tribonacci
sympy.polys.appellseqs.genocchi_poly
References
==========
.. [1] https://en.wikipedia.org/wiki/Genocchi_number
.. [2] https://mathworld.wolfram.com/GenocchiNumber.html
.. [3] Peter Luschny, "An introduction to the Bernoulli function",
https://arxiv.org/abs/2009.06743
"""
@classmethod
def eval(cls, n, x=None):
if x is S.One:
return cls(n)
elif n.is_integer is False or n.is_nonnegative is False:
return
# Genocchi numbers
elif x is None:
if n.is_odd and (n-1).is_positive:
return S.Zero
elif n.is_Number:
return 2 * (1-S(2)**n) * bernoulli(n)
# Genocchi polynomials
elif n.is_Number:
return genocchi_poly(n, x)
def _eval_rewrite_as_bernoulli(self, n, x=1, **kwargs):
if x == 1 and n.is_integer and n.is_nonnegative:
return 2 * (1-S(2)**n) * bernoulli(n)
return 2 * (bernoulli(n, x) - 2**n * bernoulli(n, (x+1) / 2))
def _eval_rewrite_as_dirichlet_eta(self, n, x=1, **kwargs):
from sympy.functions.special.zeta_functions import dirichlet_eta
return -2*n * dirichlet_eta(1-n, x)
def _eval_is_integer(self):
if len(self.args) > 1 and self.args[1] != 1:
return
n = self.args[0]
if n.is_integer and n.is_nonnegative:
return True
def _eval_is_negative(self):
if len(self.args) > 1 and self.args[1] != 1:
return
n = self.args[0]
if n.is_integer and n.is_nonnegative:
if n.is_odd:
return fuzzy_not((n-1).is_positive)
return (n/2).is_odd
def _eval_is_positive(self):
if len(self.args) > 1 and self.args[1] != 1:
return
n = self.args[0]
if n.is_integer and n.is_nonnegative:
if n.is_zero or n.is_odd:
return False
return (n/2).is_even
def _eval_is_even(self):
if len(self.args) > 1 and self.args[1] != 1:
return
n = self.args[0]
if n.is_integer and n.is_nonnegative:
if n.is_even:
return n.is_zero
return (n-1).is_positive
def _eval_is_odd(self):
if len(self.args) > 1 and self.args[1] != 1:
return
n = self.args[0]
if n.is_integer and n.is_nonnegative:
if n.is_even:
return fuzzy_not(n.is_zero)
return fuzzy_not((n-1).is_positive)
def _eval_is_prime(self):
if len(self.args) > 1 and self.args[1] != 1:
return
n = self.args[0]
# only G_6 = -3 and G_8 = 17 are prime,
# but SymPy does not consider negatives as prime
# so only n=8 is tested
return (n-8).is_zero
def _eval_evalf(self, prec):
if all(i.is_number for i in self.args):
return self.rewrite(bernoulli)._eval_evalf(prec)
#----------------------------------------------------------------------------#
# #
# Andre numbers #
# #
#----------------------------------------------------------------------------#
class andre(Function):
r"""
Andre numbers / Andre function
The Andre number `\mathcal{A}_n` is Luschny's name for half the number of
*alternating permutations* on `n` elements, where a permutation is alternating
if adjacent elements alternately compare "greater" and "smaller" going from
left to right. For example, `2 < 3 > 1 < 4` is an alternating permutation.
This sequence is A000111 in the OEIS, which assigns the names *up/down numbers*
and *Euler zigzag numbers*. It satisfies a recurrence relation similar to that
for the Catalan numbers, with `\mathcal{A}_0 = 1` and
.. math:: 2 \mathcal{A}_{n+1} = \sum_{k=0}^n \binom{n}{k} \mathcal{A}_k \mathcal{A}_{n-k}
The Bernoulli and Euler numbers are signed transformations of the odd- and
even-indexed elements of this sequence respectively:
.. math :: \operatorname{B}_{2k} = \frac{2k \mathcal{A}_{2k-1}}{(-4)^k - (-16)^k}
.. math :: \operatorname{E}_{2k} = (-1)^k \mathcal{A}_{2k}
Like the Bernoulli and Euler numbers, the Andre numbers are interpolated by the
entire Andre function:
.. math :: \mathcal{A}(s) = (-i)^{s+1} \operatorname{Li}_{-s}(i) +
i^{s+1} \operatorname{Li}_{-s}(-i) = \\ \frac{2 \Gamma(s+1)}{(2\pi)^{s+1}}
(\zeta(s+1, 1/4) - \zeta(s+1, 3/4) \cos{\pi s})
Examples
========
>>> from sympy import andre, euler, bernoulli
>>> [andre(n) for n in range(11)]
[1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, 50521]
>>> [(-1)**k * andre(2*k) for k in range(7)]
[1, -1, 5, -61, 1385, -50521, 2702765]
>>> [euler(2*k) for k in range(7)]
[1, -1, 5, -61, 1385, -50521, 2702765]
>>> [andre(2*k-1) * (2*k) / ((-4)**k - (-16)**k) for k in range(1, 8)]
[1/6, -1/30, 1/42, -1/30, 5/66, -691/2730, 7/6]
>>> [bernoulli(2*k) for k in range(1, 8)]
[1/6, -1/30, 1/42, -1/30, 5/66, -691/2730, 7/6]
See Also
========
bernoulli, catalan, euler, sympy.polys.appellseqs.andre_poly
References
==========
.. [1] https://en.wikipedia.org/wiki/Alternating_permutation
.. [2] https://mathworld.wolfram.com/EulerZigzagNumber.html
.. [3] Peter Luschny, "An introduction to the Bernoulli function",
https://arxiv.org/abs/2009.06743
"""
@classmethod
def eval(cls, n):
if n is S.NaN:
return S.NaN
elif n is S.Infinity:
return S.Infinity
if n.is_zero:
return S.One
elif n == -1:
return -log(2)
elif n == -2:
return -2*S.Catalan
elif n.is_Integer:
if n.is_nonnegative and n.is_even:
return abs(euler(n))
elif n.is_odd:
from sympy.functions.special.zeta_functions import zeta
m = -n-1
return I**m * Rational(1-2**m, 4**m) * zeta(-n)
def _eval_rewrite_as_zeta(self, s, **kwargs):
from sympy.functions.elementary.trigonometric import cos
from sympy.functions.special.gamma_functions import gamma
from sympy.functions.special.zeta_functions import zeta
return 2 * gamma(s+1) / (2*pi)**(s+1) * \
(zeta(s+1, S.One/4) - cos(pi*s) * zeta(s+1, S(3)/4))
def _eval_rewrite_as_polylog(self, s, **kwargs):
from sympy.functions.special.zeta_functions import polylog
return (-I)**(s+1) * polylog(-s, I) + I**(s+1) * polylog(-s, -I)
def _eval_is_integer(self):
n = self.args[0]
if n.is_integer and n.is_nonnegative:
return True
def _eval_is_positive(self):
if self.args[0].is_nonnegative:
return True
def _eval_evalf(self, prec):
if not self.args[0].is_number:
return
s = self.args[0]._to_mpmath(prec+12)
with workprec(prec+12):
sp, cp = mp.sinpi(s/2), mp.cospi(s/2)
res = 2*mp.dirichlet(-s, (-sp, cp, sp, -cp))
return Expr._from_mpmath(res, prec)
#----------------------------------------------------------------------------#
# #
# Partition numbers #
# #
#----------------------------------------------------------------------------#
_npartition = [1, 1]
class partition(Function):
r"""
Partition numbers
The Partition numbers are a sequence of integers `p_n` that represent the
number of distinct ways of representing `n` as a sum of natural numbers
(with order irrelevant). The generating function for `p_n` is given by:
.. math:: \sum_{n=0}^\infty p_n x^n = \prod_{k=1}^\infty (1 - x^k)^{-1}
Examples
========
>>> from sympy import partition, Symbol
>>> [partition(n) for n in range(9)]
[1, 1, 2, 3, 5, 7, 11, 15, 22]
>>> n = Symbol('n', integer=True, negative=True)
>>> partition(n)
0
See Also
========
bell, bernoulli, catalan, euler, fibonacci, harmonic, lucas, genocchi, tribonacci
References
==========
.. [1] https://en.wikipedia.org/wiki/Partition_(number_theory%29
.. [2] https://en.wikipedia.org/wiki/Pentagonal_number_theorem
"""
@staticmethod
def _partition(n):
L = len(_npartition)
if n < L:
return _npartition[n]
# lengthen cache
for _n in range(L, n + 1):
v, p, i = 0, 0, 0
while 1:
s = 0
p += 3*i + 1 # p = pentagonal number: 1, 5, 12, ...
if _n >= p:
s += _npartition[_n - p]
i += 1
gp = p + i # gp = generalized pentagonal: 2, 7, 15, ...
if _n >= gp:
s += _npartition[_n - gp]
if s == 0:
break
else:
v += s if i%2 == 1 else -s
_npartition.append(v)
return v
@classmethod
def eval(cls, n):
is_int = n.is_integer
if is_int == False:
raise ValueError("Partition numbers are defined only for "
"integers")
elif is_int:
if n.is_negative:
return S.Zero
if n.is_zero or (n - 1).is_zero:
return S.One
if n.is_Integer:
return Integer(cls._partition(n))
def _eval_is_integer(self):
if self.args[0].is_integer:
return True
def _eval_is_negative(self):
if self.args[0].is_integer:
return False
def _eval_is_positive(self):
n = self.args[0]
if n.is_nonnegative and n.is_integer:
return True
#######################################################################
###
### Functions for enumerating partitions, permutations and combinations
###
#######################################################################
class _MultisetHistogram(tuple):
pass
_N = -1
_ITEMS = -2
_M = slice(None, _ITEMS)
def _multiset_histogram(n):
"""Return tuple used in permutation and combination counting. Input
is a dictionary giving items with counts as values or a sequence of
items (which need not be sorted).
The data is stored in a class deriving from tuple so it is easily
recognized and so it can be converted easily to a list.
"""
if isinstance(n, dict): # item: count
if not all(isinstance(v, int) and v >= 0 for v in n.values()):
raise ValueError
tot = sum(n.values())
items = sum(1 for k in n if n[k] > 0)
return _MultisetHistogram([n[k] for k in n if n[k] > 0] + [items, tot])
else:
n = list(n)
s = set(n)
lens = len(s)
lenn = len(n)
if lens == lenn:
n = [1]*lenn + [lenn, lenn]
return _MultisetHistogram(n)
m = dict(zip(s, range(lens)))
d = dict(zip(range(lens), (0,)*lens))
for i in n:
d[m[i]] += 1
return _multiset_histogram(d)
def nP(n, k=None, replacement=False):
"""Return the number of permutations of ``n`` items taken ``k`` at a time.
Possible values for ``n``:
integer - set of length ``n``
sequence - converted to a multiset internally
multiset - {element: multiplicity}
If ``k`` is None then the total of all permutations of length 0
through the number of items represented by ``n`` will be returned.
If ``replacement`` is True then a given item can appear more than once
in the ``k`` items. (For example, for 'ab' permutations of 2 would
include 'aa', 'ab', 'ba' and 'bb'.) The multiplicity of elements in
``n`` is ignored when ``replacement`` is True but the total number
of elements is considered since no element can appear more times than
the number of elements in ``n``.
Examples
========
>>> from sympy.functions.combinatorial.numbers import nP
>>> from sympy.utilities.iterables import multiset_permutations, multiset
>>> nP(3, 2)
6
>>> nP('abc', 2) == nP(multiset('abc'), 2) == 6
True
>>> nP('aab', 2)
3
>>> nP([1, 2, 2], 2)
3
>>> [nP(3, i) for i in range(4)]
[1, 3, 6, 6]
>>> nP(3) == sum(_)
True
When ``replacement`` is True, each item can have multiplicity
equal to the length represented by ``n``:
>>> nP('aabc', replacement=True)
121
>>> [len(list(multiset_permutations('aaaabbbbcccc', i))) for i in range(5)]
[1, 3, 9, 27, 81]
>>> sum(_)
121
See Also
========
sympy.utilities.iterables.multiset_permutations
References
==========
.. [1] https://en.wikipedia.org/wiki/Permutation
"""
try:
n = as_int(n)
except ValueError:
return Integer(_nP(_multiset_histogram(n), k, replacement))
return Integer(_nP(n, k, replacement))
@cacheit
def _nP(n, k=None, replacement=False):
if k == 0:
return 1
if isinstance(n, SYMPY_INTS): # n different items
# assert n >= 0
if k is None:
return sum(_nP(n, i, replacement) for i in range(n + 1))
elif replacement:
return n**k
elif k > n:
return 0
elif k == n:
return factorial(k)
elif k == 1:
return n
else:
# assert k >= 0
return _product(n - k + 1, n)
elif isinstance(n, _MultisetHistogram):
if k is None:
return sum(_nP(n, i, replacement) for i in range(n[_N] + 1))
elif replacement:
return n[_ITEMS]**k
elif k == n[_N]:
return factorial(k)/prod([factorial(i) for i in n[_M] if i > 1])
elif k > n[_N]:
return 0
elif k == 1:
return n[_ITEMS]
else:
# assert k >= 0
tot = 0
n = list(n)
for i in range(len(n[_M])):
if not n[i]:
continue
n[_N] -= 1
if n[i] == 1:
n[i] = 0
n[_ITEMS] -= 1
tot += _nP(_MultisetHistogram(n), k - 1)
n[_ITEMS] += 1
n[i] = 1
else:
n[i] -= 1
tot += _nP(_MultisetHistogram(n), k - 1)
n[i] += 1
n[_N] += 1
return tot
@cacheit
def _AOP_product(n):
"""for n = (m1, m2, .., mk) return the coefficients of the polynomial,
prod(sum(x**i for i in range(nj + 1)) for nj in n); i.e. the coefficients
of the product of AOPs (all-one polynomials) or order given in n. The
resulting coefficient corresponding to x**r is the number of r-length
combinations of sum(n) elements with multiplicities given in n.
The coefficients are given as a default dictionary (so if a query is made
for a key that is not present, 0 will be returned).
Examples
========
>>> from sympy.functions.combinatorial.numbers import _AOP_product
>>> from sympy.abc import x
>>> n = (2, 2, 3) # e.g. aabbccc
>>> prod = ((x**2 + x + 1)*(x**2 + x + 1)*(x**3 + x**2 + x + 1)).expand()
>>> c = _AOP_product(n); dict(c)
{0: 1, 1: 3, 2: 6, 3: 8, 4: 8, 5: 6, 6: 3, 7: 1}
>>> [c[i] for i in range(8)] == [prod.coeff(x, i) for i in range(8)]
True
The generating poly used here is the same as that listed in
https://tinyurl.com/cep849r, but in a refactored form.
"""
n = list(n)
ord = sum(n)
need = (ord + 2)//2
rv = [1]*(n.pop() + 1)
rv.extend((0,) * (need - len(rv)))
rv = rv[:need]
while n:
ni = n.pop()
N = ni + 1
was = rv[:]
for i in range(1, min(N, len(rv))):
rv[i] += rv[i - 1]
for i in range(N, need):
rv[i] += rv[i - 1] - was[i - N]
rev = list(reversed(rv))
if ord % 2:
rv = rv + rev
else:
rv[-1:] = rev
d = defaultdict(int)
for i, r in enumerate(rv):
d[i] = r
return d
def nC(n, k=None, replacement=False):
"""Return the number of combinations of ``n`` items taken ``k`` at a time.
Possible values for ``n``:
integer - set of length ``n``
sequence - converted to a multiset internally
multiset - {element: multiplicity}
If ``k`` is None then the total of all combinations of length 0
through the number of items represented in ``n`` will be returned.
If ``replacement`` is True then a given item can appear more than once
in the ``k`` items. (For example, for 'ab' sets of 2 would include 'aa',
'ab', and 'bb'.) The multiplicity of elements in ``n`` is ignored when
``replacement`` is True but the total number of elements is considered
since no element can appear more times than the number of elements in
``n``.
Examples
========
>>> from sympy.functions.combinatorial.numbers import nC
>>> from sympy.utilities.iterables import multiset_combinations
>>> nC(3, 2)
3
>>> nC('abc', 2)
3
>>> nC('aab', 2)
2
When ``replacement`` is True, each item can have multiplicity
equal to the length represented by ``n``:
>>> nC('aabc', replacement=True)
35
>>> [len(list(multiset_combinations('aaaabbbbcccc', i))) for i in range(5)]
[1, 3, 6, 10, 15]
>>> sum(_)
35
If there are ``k`` items with multiplicities ``m_1, m_2, ..., m_k``
then the total of all combinations of length 0 through ``k`` is the
product, ``(m_1 + 1)*(m_2 + 1)*...*(m_k + 1)``. When the multiplicity
of each item is 1 (i.e., k unique items) then there are 2**k
combinations. For example, if there are 4 unique items, the total number
of combinations is 16:
>>> sum(nC(4, i) for i in range(5))
16
See Also
========
sympy.utilities.iterables.multiset_combinations
References
==========
.. [1] https://en.wikipedia.org/wiki/Combination
.. [2] https://tinyurl.com/cep849r
"""
if isinstance(n, SYMPY_INTS):
if k is None:
if not replacement:
return 2**n
return sum(nC(n, i, replacement) for i in range(n + 1))
if k < 0:
raise ValueError("k cannot be negative")
if replacement:
return binomial(n + k - 1, k)
return binomial(n, k)
if isinstance(n, _MultisetHistogram):
N = n[_N]
if k is None:
if not replacement:
return prod(m + 1 for m in n[_M])
return sum(nC(n, i, replacement) for i in range(N + 1))
elif replacement:
return nC(n[_ITEMS], k, replacement)
# assert k >= 0
elif k in (1, N - 1):
return n[_ITEMS]
elif k in (0, N):
return 1
return _AOP_product(tuple(n[_M]))[k]
else:
return nC(_multiset_histogram(n), k, replacement)
def _eval_stirling1(n, k):
if n == k == 0:
return S.One
if 0 in (n, k):
return S.Zero
# some special values
if n == k:
return S.One
elif k == n - 1:
return binomial(n, 2)
elif k == n - 2:
return (3*n - 1)*binomial(n, 3)/4
elif k == n - 3:
return binomial(n, 2)*binomial(n, 4)
return _stirling1(n, k)
@cacheit
def _stirling1(n, k):
row = [0, 1]+[0]*(k-1) # for n = 1
for i in range(2, n+1):
for j in range(min(k,i), 0, -1):
row[j] = (i-1) * row[j] + row[j-1]
return Integer(row[k])
def _eval_stirling2(n, k):
if n == k == 0:
return S.One
if 0 in (n, k):
return S.Zero
# some special values
if n == k:
return S.One
elif k == n - 1:
return binomial(n, 2)
elif k == 1:
return S.One
elif k == 2:
return Integer(2**(n - 1) - 1)
return _stirling2(n, k)
@cacheit
def _stirling2(n, k):
row = [0, 1]+[0]*(k-1) # for n = 1
for i in range(2, n+1):
for j in range(min(k,i), 0, -1):
row[j] = j * row[j] + row[j-1]
return Integer(row[k])
def stirling(n, k, d=None, kind=2, signed=False):
r"""Return Stirling number $S(n, k)$ of the first or second (default) kind.
The sum of all Stirling numbers of the second kind for $k = 1$
through $n$ is ``bell(n)``. The recurrence relationship for these numbers
is:
.. math :: {0 \brace 0} = 1; {n \brace 0} = {0 \brace k} = 0;
.. math :: {{n+1} \brace k} = j {n \brace k} + {n \brace {k-1}}
where $j$ is:
$n$ for Stirling numbers of the first kind,
$-n$ for signed Stirling numbers of the first kind,
$k$ for Stirling numbers of the second kind.
The first kind of Stirling number counts the number of permutations of
``n`` distinct items that have ``k`` cycles; the second kind counts the
ways in which ``n`` distinct items can be partitioned into ``k`` parts.
If ``d`` is given, the "reduced Stirling number of the second kind" is
returned: $S^{d}(n, k) = S(n - d + 1, k - d + 1)$ with $n \ge k \ge d$.
(This counts the ways to partition $n$ consecutive integers into $k$
groups with no pairwise difference less than $d$. See example below.)
To obtain the signed Stirling numbers of the first kind, use keyword
``signed=True``. Using this keyword automatically sets ``kind`` to 1.
Examples
========
>>> from sympy.functions.combinatorial.numbers import stirling, bell
>>> from sympy.combinatorics import Permutation
>>> from sympy.utilities.iterables import multiset_partitions, permutations
First kind (unsigned by default):
>>> [stirling(6, i, kind=1) for i in range(7)]
[0, 120, 274, 225, 85, 15, 1]
>>> perms = list(permutations(range(4)))
>>> [sum(Permutation(p).cycles == i for p in perms) for i in range(5)]
[0, 6, 11, 6, 1]
>>> [stirling(4, i, kind=1) for i in range(5)]
[0, 6, 11, 6, 1]
First kind (signed):
>>> [stirling(4, i, signed=True) for i in range(5)]
[0, -6, 11, -6, 1]
Second kind:
>>> [stirling(10, i) for i in range(12)]
[0, 1, 511, 9330, 34105, 42525, 22827, 5880, 750, 45, 1, 0]
>>> sum(_) == bell(10)
True
>>> len(list(multiset_partitions(range(4), 2))) == stirling(4, 2)
True
Reduced second kind:
>>> from sympy import subsets, oo
>>> def delta(p):
... if len(p) == 1:
... return oo
... return min(abs(i[0] - i[1]) for i in subsets(p, 2))
>>> parts = multiset_partitions(range(5), 3)
>>> d = 2
>>> sum(1 for p in parts if all(delta(i) >= d for i in p))
7
>>> stirling(5, 3, 2)
7
See Also
========
sympy.utilities.iterables.multiset_partitions
References
==========
.. [1] https://en.wikipedia.org/wiki/Stirling_numbers_of_the_first_kind
.. [2] https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind
"""
# TODO: make this a class like bell()
n = as_int(n)
k = as_int(k)
if n < 0:
raise ValueError('n must be nonnegative')
if k > n:
return S.Zero
if d:
# assert k >= d
# kind is ignored -- only kind=2 is supported
return _eval_stirling2(n - d + 1, k - d + 1)
elif signed:
# kind is ignored -- only kind=1 is supported
return S.NegativeOne**(n - k)*_eval_stirling1(n, k)
if kind == 1:
return _eval_stirling1(n, k)
elif kind == 2:
return _eval_stirling2(n, k)
else:
raise ValueError('kind must be 1 or 2, not %s' % k)
@cacheit
def _nT(n, k):
"""Return the partitions of ``n`` items into ``k`` parts. This
is used by ``nT`` for the case when ``n`` is an integer."""
# really quick exits
if k > n or k < 0:
return 0
if k in (1, n):
return 1
if k == 0:
return 0
# exits that could be done below but this is quicker
if k == 2:
return n//2
d = n - k
if d <= 3:
return d
# quick exit
if 3*k >= n: # or, equivalently, 2*k >= d
# all the information needed in this case
# will be in the cache needed to calculate
# partition(d), so...
# update cache
tot = partition._partition(d)
# and correct for values not needed
if d - k > 0:
tot -= sum(_npartition[:d - k])
return tot
# regular exit
# nT(n, k) = Sum(nT(n - k, m), (m, 1, k));
# calculate needed nT(i, j) values
p = [1]*d
for i in range(2, k + 1):
for m in range(i + 1, d):
p[m] += p[m - i]
d -= 1
# if p[0] were appended to the end of p then the last
# k values of p are the nT(n, j) values for 0 < j < k in reverse
# order p[-1] = nT(n, 1), p[-2] = nT(n, 2), etc.... Instead of
# putting the 1 from p[0] there, however, it is simply added to
# the sum below which is valid for 1 < k <= n//2
return (1 + sum(p[1 - k:]))
def nT(n, k=None):
"""Return the number of ``k``-sized partitions of ``n`` items.
Possible values for ``n``:
integer - ``n`` identical items
sequence - converted to a multiset internally
multiset - {element: multiplicity}
Note: the convention for ``nT`` is different than that of ``nC`` and
``nP`` in that
here an integer indicates ``n`` *identical* items instead of a set of
length ``n``; this is in keeping with the ``partitions`` function which
treats its integer-``n`` input like a list of ``n`` 1s. One can use
``range(n)`` for ``n`` to indicate ``n`` distinct items.
If ``k`` is None then the total number of ways to partition the elements
represented in ``n`` will be returned.
Examples
========
>>> from sympy.functions.combinatorial.numbers import nT
Partitions of the given multiset:
>>> [nT('aabbc', i) for i in range(1, 7)]
[1, 8, 11, 5, 1, 0]
>>> nT('aabbc') == sum(_)
True
>>> [nT("mississippi", i) for i in range(1, 12)]
[1, 74, 609, 1521, 1768, 1224, 579, 197, 50, 9, 1]
Partitions when all items are identical:
>>> [nT(5, i) for i in range(1, 6)]
[1, 2, 2, 1, 1]
>>> nT('1'*5) == sum(_)
True
When all items are different:
>>> [nT(range(5), i) for i in range(1, 6)]
[1, 15, 25, 10, 1]
>>> nT(range(5)) == sum(_)
True
Partitions of an integer expressed as a sum of positive integers:
>>> from sympy import partition
>>> partition(4)
5
>>> nT(4, 1) + nT(4, 2) + nT(4, 3) + nT(4, 4)
5
>>> nT('1'*4)
5
See Also
========
sympy.utilities.iterables.partitions
sympy.utilities.iterables.multiset_partitions
sympy.functions.combinatorial.numbers.partition
References
==========
.. [1] https://web.archive.org/web/20210507012732/https://teaching.csse.uwa.edu.au/units/CITS7209/partition.pdf
"""
if isinstance(n, SYMPY_INTS):
# n identical items
if k is None:
return partition(n)
if isinstance(k, SYMPY_INTS):
n = as_int(n)
k = as_int(k)
return Integer(_nT(n, k))
if not isinstance(n, _MultisetHistogram):
try:
# if n contains hashable items there is some
# quick handling that can be done
u = len(set(n))
if u <= 1:
return nT(len(n), k)
elif u == len(n):
n = range(u)
raise TypeError
except TypeError:
n = _multiset_histogram(n)
N = n[_N]
if k is None and N == 1:
return 1
if k in (1, N):
return 1
if k == 2 or N == 2 and k is None:
m, r = divmod(N, 2)
rv = sum(nC(n, i) for i in range(1, m + 1))
if not r:
rv -= nC(n, m)//2
if k is None:
rv += 1 # for k == 1
return rv
if N == n[_ITEMS]:
# all distinct
if k is None:
return bell(N)
return stirling(N, k)
m = MultisetPartitionTraverser()
if k is None:
return m.count_partitions(n[_M])
# MultisetPartitionTraverser does not have a range-limited count
# method, so need to enumerate and count
tot = 0
for discard in m.enum_range(n[_M], k-1, k):
tot += 1
return tot
#-----------------------------------------------------------------------------#
# #
# Motzkin numbers #
# #
#-----------------------------------------------------------------------------#
class motzkin(Function):
"""
The nth Motzkin number is the number
of ways of drawing non-intersecting chords
between n points on a circle (not necessarily touching
every point by a chord). The Motzkin numbers are named
after Theodore Motzkin and have diverse applications
in geometry, combinatorics and number theory.
Motzkin numbers are the integer sequence defined by the
initial terms `M_0 = 1`, `M_1 = 1` and the two-term recurrence relation
`M_n = \frac{2*n + 1}{n + 2} * M_{n-1} + \frac{3n - 3}{n + 2} * M_{n-2}`.
Examples
========
>>> from sympy import motzkin
>>> motzkin.is_motzkin(5)
False
>>> motzkin.find_motzkin_numbers_in_range(2,300)
[2, 4, 9, 21, 51, 127]
>>> motzkin.find_motzkin_numbers_in_range(2,900)
[2, 4, 9, 21, 51, 127, 323, 835]
>>> motzkin.find_first_n_motzkins(10)
[1, 1, 2, 4, 9, 21, 51, 127, 323, 835]
References
==========
.. [1] https://en.wikipedia.org/wiki/Motzkin_number
.. [2] https://mathworld.wolfram.com/MotzkinNumber.html
"""
@staticmethod
def is_motzkin(n):
try:
n = as_int(n)
except ValueError:
return False
if n > 0:
if n in (1, 2):
return True
tn1 = 1
tn = 2
i = 3
while tn < n:
a = ((2*i + 1)*tn + (3*i - 3)*tn1)/(i + 2)
i += 1
tn1 = tn
tn = a
if tn == n:
return True
else:
return False
else:
return False
@staticmethod
def find_motzkin_numbers_in_range(x, y):
if 0 <= x <= y:
motzkins = []
if x <= 1 <= y:
motzkins.append(1)
tn1 = 1
tn = 2
i = 3
while tn <= y:
if tn >= x:
motzkins.append(tn)
a = ((2*i + 1)*tn + (3*i - 3)*tn1)/(i + 2)
i += 1
tn1 = tn
tn = int(a)
return motzkins
else:
raise ValueError('The provided range is not valid. This condition should satisfy x <= y')
@staticmethod
def find_first_n_motzkins(n):
try:
n = as_int(n)
except ValueError:
raise ValueError('The provided number must be a positive integer')
if n < 0:
raise ValueError('The provided number must be a positive integer')
motzkins = [1]
if n >= 1:
motzkins.append(1)
tn1 = 1
tn = 2
i = 3
while i <= n:
motzkins.append(tn)
a = ((2*i + 1)*tn + (3*i - 3)*tn1)/(i + 2)
i += 1
tn1 = tn
tn = int(a)
return motzkins
@staticmethod
@recurrence_memo([S.One, S.One])
def _motzkin(n, prev):
return ((2*n + 1)*prev[-1] + (3*n - 3)*prev[-2]) // (n + 2)
@classmethod
def eval(cls, n):
try:
n = as_int(n)
except ValueError:
raise ValueError('The provided number must be a positive integer')
if n < 0:
raise ValueError('The provided number must be a positive integer')
return Integer(cls._motzkin(n - 1))
def nD(i=None, brute=None, *, n=None, m=None):
"""return the number of derangements for: ``n`` unique items, ``i``
items (as a sequence or multiset), or multiplicities, ``m`` given
as a sequence or multiset.
Examples
========
>>> from sympy.utilities.iterables import generate_derangements as enum
>>> from sympy.functions.combinatorial.numbers import nD
A derangement ``d`` of sequence ``s`` has all ``d[i] != s[i]``:
>>> set([''.join(i) for i in enum('abc')])
{'bca', 'cab'}
>>> nD('abc')
2
Input as iterable or dictionary (multiset form) is accepted:
>>> assert nD([1, 2, 2, 3, 3, 3]) == nD({1: 1, 2: 2, 3: 3})
By default, a brute-force enumeration and count of multiset permutations
is only done if there are fewer than 9 elements. There may be cases when
there is high multiplicity with few unique elements that will benefit
from a brute-force enumeration, too. For this reason, the `brute`
keyword (default None) is provided. When False, the brute-force
enumeration will never be used. When True, it will always be used.
>>> nD('1111222233', brute=True)
44
For convenience, one may specify ``n`` distinct items using the
``n`` keyword:
>>> assert nD(n=3) == nD('abc') == 2
Since the number of derangments depends on the multiplicity of the
elements and not the elements themselves, it may be more convenient
to give a list or multiset of multiplicities using keyword ``m``:
>>> assert nD('abc') == nD(m=(1,1,1)) == nD(m={1:3}) == 2
"""
from sympy.integrals.integrals import integrate
from sympy.functions.special.polynomials import laguerre
from sympy.abc import x
def ok(x):
if not isinstance(x, SYMPY_INTS):
raise TypeError('expecting integer values')
if x < 0:
raise ValueError('value must not be negative')
return True
if (i, n, m).count(None) != 2:
raise ValueError('enter only 1 of i, n, or m')
if i is not None:
if isinstance(i, SYMPY_INTS):
raise TypeError('items must be a list or dictionary')
if not i:
return S.Zero
if type(i) is not dict:
s = list(i)
ms = multiset(s)
elif type(i) is dict:
all(ok(_) for _ in i.values())
ms = {k: v for k, v in i.items() if v}
s = None
if not ms:
return S.Zero
N = sum(ms.values())
counts = multiset(ms.values())
nkey = len(ms)
elif n is not None:
ok(n)
if not n:
return S.Zero
return subfactorial(n)
elif m is not None:
if isinstance(m, dict):
all(ok(i) and ok(j) for i, j in m.items())
counts = {k: v for k, v in m.items() if k*v}
elif iterable(m) or isinstance(m, str):
m = list(m)
all(ok(i) for i in m)
counts = multiset([i for i in m if i])
else:
raise TypeError('expecting iterable')
if not counts:
return S.Zero
N = sum(k*v for k, v in counts.items())
nkey = sum(counts.values())
s = None
big = int(max(counts))
if big == 1: # no repetition
return subfactorial(nkey)
nval = len(counts)
if big*2 > N:
return S.Zero
if big*2 == N:
if nkey == 2 and nval == 1:
return S.One # aaabbb
if nkey - 1 == big: # one element repeated
return factorial(big) # e.g. abc part of abcddd
if N < 9 and brute is None or brute:
# for all possibilities, this was found to be faster
if s is None:
s = []
i = 0
for m, v in counts.items():
for j in range(v):
s.extend([i]*m)
i += 1
return Integer(sum(1 for i in multiset_derangements(s)))
from sympy.functions.elementary.exponential import exp
return Integer(abs(integrate(exp(-x)*Mul(*[
laguerre(i, x)**m for i, m in counts.items()]), (x, 0, oo))))
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