from __future__ import annotations

from copy import deepcopy
from functools import reduce

from sympy.core import S, sympify, Dummy, Mod
from sympy.core.cache import cacheit
from sympy.core.function import Function, ArgumentIndexError, PoleError
from sympy.core.logic import fuzzy_and
from sympy.core.numbers import Integer, pi, I
from sympy.core.relational import Eq
from sympy.external.gmpy import HAS_GMPY, gmpy
from sympy.ntheory import sieve
from sympy.ntheory.residue_ntheory import binomial_mod
from sympy.polys.polytools import Poly

from math import factorial as _factorial, prod, sqrt as _sqrt

class CombinatorialFunction(Function):
    """Base class for combinatorial functions. """

    def _eval_simplify(self, **kwargs):
        from sympy.simplify.combsimp import combsimp
        # combinatorial function with non-integer arguments is
        # automatically passed to gammasimp
        expr = combsimp(self)
        measure = kwargs['measure']
        if measure(expr) <= kwargs['ratio']*measure(self):
            return expr
        return self


###############################################################################
######################## FACTORIAL and MULTI-FACTORIAL ########################
###############################################################################


class factorial(CombinatorialFunction):
    r"""Implementation of factorial function over nonnegative integers.
       By convention (consistent with the gamma function and the binomial
       coefficients), factorial of a negative integer is complex infinity.

       The factorial is very important in combinatorics where it gives
       the number of ways in which `n` objects can be permuted. It also
       arises in calculus, probability, number theory, etc.

       There is strict relation of factorial with gamma function. In
       fact `n! = gamma(n+1)` for nonnegative integers. Rewrite of this
       kind is very useful in case of combinatorial simplification.

       Computation of the factorial is done using two algorithms. For
       small arguments a precomputed look up table is used. However for bigger
       input algorithm Prime-Swing is used. It is the fastest algorithm
       known and computes `n!` via prime factorization of special class
       of numbers, called here the 'Swing Numbers'.

       Examples
       ========

       >>> from sympy import Symbol, factorial, S
       >>> n = Symbol('n', integer=True)

       >>> factorial(0)
       1

       >>> factorial(7)
       5040

       >>> factorial(-2)
       zoo

       >>> factorial(n)
       factorial(n)

       >>> factorial(2*n)
       factorial(2*n)

       >>> factorial(S(1)/2)
       factorial(1/2)

       See Also
       ========

       factorial2, RisingFactorial, FallingFactorial
    """

    def fdiff(self, argindex=1):
        from sympy.functions.special.gamma_functions import (gamma, polygamma)
        if argindex == 1:
            return gamma(self.args[0] + 1)*polygamma(0, self.args[0] + 1)
        else:
            raise ArgumentIndexError(self, argindex)

    _small_swing = [
        1, 1, 1, 3, 3, 15, 5, 35, 35, 315, 63, 693, 231, 3003, 429, 6435, 6435, 109395,
        12155, 230945, 46189, 969969, 88179, 2028117, 676039, 16900975, 1300075,
        35102025, 5014575, 145422675, 9694845, 300540195, 300540195
    ]

    _small_factorials: list[int] = []

    @classmethod
    def _swing(cls, n):
        if n < 33:
            return cls._small_swing[n]
        else:
            N, primes = int(_sqrt(n)), []

            for prime in sieve.primerange(3, N + 1):
                p, q = 1, n

                while True:
                    q //= prime

                    if q > 0:
                        if q & 1 == 1:
                            p *= prime
                    else:
                        break

                if p > 1:
                    primes.append(p)

            for prime in sieve.primerange(N + 1, n//3 + 1):
                if (n // prime) & 1 == 1:
                    primes.append(prime)

            L_product = prod(sieve.primerange(n//2 + 1, n + 1))
            R_product = prod(primes)

            return L_product*R_product

    @classmethod
    def _recursive(cls, n):
        if n < 2:
            return 1
        else:
            return (cls._recursive(n//2)**2)*cls._swing(n)

    @classmethod
    def eval(cls, n):
        n = sympify(n)

        if n.is_Number:
            if n.is_zero:
                return S.One
            elif n is S.Infinity:
                return S.Infinity
            elif n.is_Integer:
                if n.is_negative:
                    return S.ComplexInfinity
                else:
                    n = n.p

                    if n < 20:
                        if not cls._small_factorials:
                            result = 1
                            for i in range(1, 20):
                                result *= i
                                cls._small_factorials.append(result)
                        result = cls._small_factorials[n-1]

                    # GMPY factorial is faster, use it when available
                    elif HAS_GMPY:
                        result = gmpy.fac(n)

                    else:
                        bits = bin(n).count('1')
                        result = cls._recursive(n)*2**(n - bits)

                    return Integer(result)

    def _facmod(self, n, q):
        res, N = 1, int(_sqrt(n))

        # Exponent of prime p in n! is e_p(n) = [n/p] + [n/p**2] + ...
        # for p > sqrt(n), e_p(n) < sqrt(n), the primes with [n/p] = m,
        # occur consecutively and are grouped together in pw[m] for
        # simultaneous exponentiation at a later stage
        pw = [1]*N

        m = 2 # to initialize the if condition below
        for prime in sieve.primerange(2, n + 1):
            if m > 1:
                m, y = 0, n // prime
                while y:
                    m += y
                    y //= prime
            if m < N:
                pw[m] = pw[m]*prime % q
            else:
                res = res*pow(prime, m, q) % q

        for ex, bs in enumerate(pw):
            if ex == 0 or bs == 1:
                continue
            if bs == 0:
                return 0
            res = res*pow(bs, ex, q) % q

        return res

    def _eval_Mod(self, q):
        n = self.args[0]
        if n.is_integer and n.is_nonnegative and q.is_integer:
            aq = abs(q)
            d = aq - n
            if d.is_nonpositive:
                return S.Zero
            else:
                isprime = aq.is_prime
                if d == 1:
                    # Apply Wilson's theorem (if a natural number n > 1
                    # is a prime number, then (n-1)! = -1 mod n) and
                    # its inverse (if n > 4 is a composite number, then
                    # (n-1)! = 0 mod n)
                    if isprime:
                        return -1 % q
                    elif isprime is False and (aq - 6).is_nonnegative:
                        return S.Zero
                elif n.is_Integer and q.is_Integer:
                    n, d, aq = map(int, (n, d, aq))
                    if isprime and (d - 1 < n):
                        fc = self._facmod(d - 1, aq)
                        fc = pow(fc, aq - 2, aq)
                        if d%2:
                            fc = -fc
                    else:
                        fc = self._facmod(n, aq)

                    return fc % q

    def _eval_rewrite_as_gamma(self, n, piecewise=True, **kwargs):
        from sympy.functions.special.gamma_functions import gamma
        return gamma(n + 1)

    def _eval_rewrite_as_Product(self, n, **kwargs):
        from sympy.concrete.products import Product
        if n.is_nonnegative and n.is_integer:
            i = Dummy('i', integer=True)
            return Product(i, (i, 1, 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_integer and self.args[0].is_nonnegative:
            return True

    def _eval_is_even(self):
        x = self.args[0]
        if x.is_integer and x.is_nonnegative:
            return (x - 2).is_nonnegative

    def _eval_is_composite(self):
        x = self.args[0]
        if x.is_integer and x.is_nonnegative:
            return (x - 3).is_nonnegative

    def _eval_is_real(self):
        x = self.args[0]
        if x.is_nonnegative or x.is_noninteger:
            return True

    def _eval_as_leading_term(self, x, logx=None, cdir=0):
        arg = self.args[0].as_leading_term(x)
        arg0 = arg.subs(x, 0)
        if arg0.is_zero:
            return S.One
        elif not arg0.is_infinite:
            return self.func(arg)
        raise PoleError("Cannot expand %s around 0" % (self))

    def __deepcopy__(self, memodict={}):
        args = [deepcopy(a) for a in self.args]
        new_instance = factorial(*args, evaluate=False)
        return new_instance

class MultiFactorial(CombinatorialFunction):
    pass


class subfactorial(CombinatorialFunction):
    r"""The subfactorial counts the derangements of $n$ items and is
    defined for non-negative integers as:

    .. math:: !n = \begin{cases} 1 & n = 0 \\ 0 & n = 1 \\
                    (n-1)(!(n-1) + !(n-2)) & n > 1 \end{cases}

    It can also be written as ``int(round(n!/exp(1)))`` but the
    recursive definition with caching is implemented for this function.

    An interesting analytic expression is the following [2]_

    .. math:: !x = \Gamma(x + 1, -1)/e

    which is valid for non-negative integers `x`. The above formula
    is not very useful in case of non-integers. `\Gamma(x + 1, -1)` is
    single-valued only for integral arguments `x`, elsewhere on the positive
    real axis it has an infinite number of branches none of which are real.

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Subfactorial
    .. [2] https://mathworld.wolfram.com/Subfactorial.html

    Examples
    ========

    >>> from sympy import subfactorial
    >>> from sympy.abc import n
    >>> subfactorial(n + 1)
    subfactorial(n + 1)
    >>> subfactorial(5)
    44

    See Also
    ========

    factorial, uppergamma,
    sympy.utilities.iterables.generate_derangements
    """

    @classmethod
    @cacheit
    def _eval(self, n):
        if not n:
            return S.One
        elif n == 1:
            return S.Zero
        else:
            z1, z2 = 1, 0
            for i in range(2, n + 1):
                z1, z2 = z2, (i - 1)*(z2 + z1)
            return z2

    @classmethod
    def eval(cls, arg):
        if arg.is_Number:
            if arg.is_Integer and arg.is_nonnegative:
                return cls._eval(arg)
            elif arg is S.NaN:
                return S.NaN
            elif arg is S.Infinity:
                return S.Infinity

    def _eval_is_even(self):
        if self.args[0].is_odd and self.args[0].is_nonnegative:
            return True

    def _eval_is_integer(self):
        if self.args[0].is_integer and self.args[0].is_nonnegative:
            return True

    def _eval_rewrite_as_factorial(self, arg, **kwargs):
        from sympy.concrete.summations import summation
        i = Dummy('i')
        f = S.NegativeOne**i / factorial(i)
        return factorial(arg) * summation(f, (i, 0, arg))

    def _eval_rewrite_as_gamma(self, arg, piecewise=True, **kwargs):
        from sympy.functions.elementary.exponential import exp
        from sympy.functions.special.gamma_functions import (gamma, lowergamma)
        return (S.NegativeOne**(arg + 1)*exp(-I*pi*arg)*lowergamma(arg + 1, -1)
                + gamma(arg + 1))*exp(-1)

    def _eval_rewrite_as_uppergamma(self, arg, **kwargs):
        from sympy.functions.special.gamma_functions import uppergamma
        return uppergamma(arg + 1, -1)/S.Exp1

    def _eval_is_nonnegative(self):
        if self.args[0].is_integer and self.args[0].is_nonnegative:
            return True

    def _eval_is_odd(self):
        if self.args[0].is_even and self.args[0].is_nonnegative:
            return True


class factorial2(CombinatorialFunction):
    r"""The double factorial `n!!`, not to be confused with `(n!)!`

    The double factorial is defined for nonnegative integers and for odd
    negative integers as:

    .. math:: n!! = \begin{cases} 1 & n = 0 \\
                    n(n-2)(n-4) \cdots 1 & n\ \text{positive odd} \\
                    n(n-2)(n-4) \cdots 2 & n\ \text{positive even} \\
                    (n+2)!!/(n+2) & n\ \text{negative odd} \end{cases}

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Double_factorial

    Examples
    ========

    >>> from sympy import factorial2, var
    >>> n = var('n')
    >>> n
    n
    >>> factorial2(n + 1)
    factorial2(n + 1)
    >>> factorial2(5)
    15
    >>> factorial2(-1)
    1
    >>> factorial2(-5)
    1/3

    See Also
    ========

    factorial, RisingFactorial, FallingFactorial
    """

    @classmethod
    def eval(cls, arg):
        # TODO: extend this to complex numbers?

        if arg.is_Number:
            if not arg.is_Integer:
                raise ValueError("argument must be nonnegative integer "
                                    "or negative odd integer")

            # This implementation is faster than the recursive one
            # It also avoids "maximum recursion depth exceeded" runtime error
            if arg.is_nonnegative:
                if arg.is_even:
                    k = arg / 2
                    return 2**k * factorial(k)
                return factorial(arg) / factorial2(arg - 1)


            if arg.is_odd:
                return arg*(S.NegativeOne)**((1 - arg)/2) / factorial2(-arg)
            raise ValueError("argument must be nonnegative integer "
                                "or negative odd integer")


    def _eval_is_even(self):
        # Double factorial is even for every positive even input
        n = self.args[0]
        if n.is_integer:
            if n.is_odd:
                return False
            if n.is_even:
                if n.is_positive:
                    return True
                if n.is_zero:
                    return False

    def _eval_is_integer(self):
        # Double factorial is an integer for every nonnegative input, and for
        # -1 and -3
        n = self.args[0]
        if n.is_integer:
            if (n + 1).is_nonnegative:
                return True
            if n.is_odd:
                return (n + 3).is_nonnegative

    def _eval_is_odd(self):
        # Double factorial is odd for every odd input not smaller than -3, and
        # for 0
        n = self.args[0]
        if n.is_odd:
            return (n + 3).is_nonnegative
        if n.is_even:
            if n.is_positive:
                return False
            if n.is_zero:
                return True

    def _eval_is_positive(self):
        # Double factorial is positive for every nonnegative input, and for
        # every odd negative input which is of the form -1-4k for an
        # nonnegative integer k
        n = self.args[0]
        if n.is_integer:
            if (n + 1).is_nonnegative:
                return True
            if n.is_odd:
                return ((n + 1) / 2).is_even

    def _eval_rewrite_as_gamma(self, n, piecewise=True, **kwargs):
        from sympy.functions.elementary.miscellaneous import sqrt
        from sympy.functions.elementary.piecewise import Piecewise
        from sympy.functions.special.gamma_functions import gamma
        return 2**(n/2)*gamma(n/2 + 1) * Piecewise((1, Eq(Mod(n, 2), 0)),
                (sqrt(2/pi), Eq(Mod(n, 2), 1)))


###############################################################################
######################## RISING and FALLING FACTORIALS ########################
###############################################################################


class RisingFactorial(CombinatorialFunction):
    r"""
    Rising factorial (also called Pochhammer symbol [1]_) is a double valued
    function arising in concrete mathematics, hypergeometric functions
    and series expansions. It is defined by:

    .. math:: \texttt{rf(y, k)} = (x)^k = x \cdot (x+1) \cdots (x+k-1)

    where `x` can be arbitrary expression and `k` is an integer. For
    more information check "Concrete mathematics" by Graham, pp. 66
    or visit https://mathworld.wolfram.com/RisingFactorial.html page.

    When `x` is a `~.Poly` instance of degree $\ge 1$ with a single variable,
    `(x)^k = x(y) \cdot x(y+1) \cdots x(y+k-1)`, where `y` is the
    variable of `x`. This is as described in [2]_.

    Examples
    ========

    >>> from sympy import rf, Poly
    >>> from sympy.abc import x
    >>> rf(x, 0)
    1
    >>> rf(1, 5)
    120
    >>> rf(x, 5) == x*(1 + x)*(2 + x)*(3 + x)*(4 + x)
    True
    >>> rf(Poly(x**3, x), 2)
    Poly(x**6 + 3*x**5 + 3*x**4 + x**3, x, domain='ZZ')

    Rewriting is complicated unless the relationship between
    the arguments is known, but rising factorial can
    be rewritten in terms of gamma, factorial, binomial,
    and falling factorial.

    >>> from sympy import Symbol, factorial, ff, binomial, gamma
    >>> n = Symbol('n', integer=True, positive=True)
    >>> R = rf(n, n + 2)
    >>> for i in (rf, ff, factorial, binomial, gamma):
    ...  R.rewrite(i)
    ...
    RisingFactorial(n, n + 2)
    FallingFactorial(2*n + 1, n + 2)
    factorial(2*n + 1)/factorial(n - 1)
    binomial(2*n + 1, n + 2)*factorial(n + 2)
    gamma(2*n + 2)/gamma(n)

    See Also
    ========

    factorial, factorial2, FallingFactorial

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Pochhammer_symbol
    .. [2] Peter Paule, "Greatest Factorial Factorization and Symbolic
           Summation", Journal of Symbolic Computation, vol. 20, pp. 235-268,
           1995.

    """

    @classmethod
    def eval(cls, x, k):
        x = sympify(x)
        k = sympify(k)

        if x is S.NaN or k is S.NaN:
            return S.NaN
        elif x is S.One:
            return factorial(k)
        elif k.is_Integer:
            if k.is_zero:
                return S.One
            else:
                if k.is_positive:
                    if x is S.Infinity:
                        return S.Infinity
                    elif x is S.NegativeInfinity:
                        if k.is_odd:
                            return S.NegativeInfinity
                        else:
                            return S.Infinity
                    else:
                        if isinstance(x, Poly):
                            gens = x.gens
                            if len(gens)!= 1:
                                raise ValueError("rf only defined for "
                                            "polynomials on one generator")
                            else:
                                return reduce(lambda r, i:
                                              r*(x.shift(i)),
                                              range(int(k)), 1)
                        else:
                            return reduce(lambda r, i: r*(x + i),
                                          range(int(k)), 1)

                else:
                    if x is S.Infinity:
                        return S.Infinity
                    elif x is S.NegativeInfinity:
                        return S.Infinity
                    else:
                        if isinstance(x, Poly):
                            gens = x.gens
                            if len(gens)!= 1:
                                raise ValueError("rf only defined for "
                                            "polynomials on one generator")
                            else:
                                return 1/reduce(lambda r, i:
                                                r*(x.shift(-i)),
                                                range(1, abs(int(k)) + 1), 1)
                        else:
                            return 1/reduce(lambda r, i:
                                            r*(x - i),
                                            range(1, abs(int(k)) + 1), 1)

        if k.is_integer == False:
            if x.is_integer and x.is_negative:
                return S.Zero

    def _eval_rewrite_as_gamma(self, x, k, piecewise=True, **kwargs):
        from sympy.functions.elementary.piecewise import Piecewise
        from sympy.functions.special.gamma_functions import gamma
        if not piecewise:
            if (x <= 0) == True:
                return S.NegativeOne**k*gamma(1 - x) / gamma(-k - x + 1)
            return gamma(x + k) / gamma(x)
        return Piecewise(
            (gamma(x + k) / gamma(x), x > 0),
            (S.NegativeOne**k*gamma(1 - x) / gamma(-k - x + 1), True))

    def _eval_rewrite_as_FallingFactorial(self, x, k, **kwargs):
        return FallingFactorial(x + k - 1, k)

    def _eval_rewrite_as_factorial(self, x, k, **kwargs):
        from sympy.functions.elementary.piecewise import Piecewise
        if x.is_integer and k.is_integer:
            return Piecewise(
                (factorial(k + x - 1)/factorial(x - 1), x > 0),
                (S.NegativeOne**k*factorial(-x)/factorial(-k - x), True))

    def _eval_rewrite_as_binomial(self, x, k, **kwargs):
        if k.is_integer:
            return factorial(k) * binomial(x + k - 1, k)

    def _eval_rewrite_as_tractable(self, x, k, limitvar=None, **kwargs):
        from sympy.functions.special.gamma_functions import gamma
        if limitvar:
            k_lim = k.subs(limitvar, S.Infinity)
            if k_lim is S.Infinity:
                return (gamma(x + k).rewrite('tractable', deep=True) / gamma(x))
            elif k_lim is S.NegativeInfinity:
                return (S.NegativeOne**k*gamma(1 - x) / gamma(-k - x + 1).rewrite('tractable', deep=True))
        return self.rewrite(gamma).rewrite('tractable', deep=True)

    def _eval_is_integer(self):
        return fuzzy_and((self.args[0].is_integer, self.args[1].is_integer,
                          self.args[1].is_nonnegative))


class FallingFactorial(CombinatorialFunction):
    r"""
    Falling factorial (related to rising factorial) is a double valued
    function arising in concrete mathematics, hypergeometric functions
    and series expansions. It is defined by

    .. math:: \texttt{ff(x, k)} = (x)_k = x \cdot (x-1) \cdots (x-k+1)

    where `x` can be arbitrary expression and `k` is an integer. For
    more information check "Concrete mathematics" by Graham, pp. 66
    or [1]_.

    When `x` is a `~.Poly` instance of degree $\ge 1$ with single variable,
    `(x)_k = x(y) \cdot x(y-1) \cdots x(y-k+1)`, where `y` is the
    variable of `x`. This is as described in

    >>> from sympy import ff, Poly, Symbol
    >>> from sympy.abc import x
    >>> n = Symbol('n', integer=True)

    >>> ff(x, 0)
    1
    >>> ff(5, 5)
    120
    >>> ff(x, 5) == x*(x - 1)*(x - 2)*(x - 3)*(x - 4)
    True
    >>> ff(Poly(x**2, x), 2)
    Poly(x**4 - 2*x**3 + x**2, x, domain='ZZ')
    >>> ff(n, n)
    factorial(n)

    Rewriting is complicated unless the relationship between
    the arguments is known, but falling factorial can
    be rewritten in terms of gamma, factorial and binomial
    and rising factorial.

    >>> from sympy import factorial, rf, gamma, binomial, Symbol
    >>> n = Symbol('n', integer=True, positive=True)
    >>> F = ff(n, n - 2)
    >>> for i in (rf, ff, factorial, binomial, gamma):
    ...  F.rewrite(i)
    ...
    RisingFactorial(3, n - 2)
    FallingFactorial(n, n - 2)
    factorial(n)/2
    binomial(n, n - 2)*factorial(n - 2)
    gamma(n + 1)/2

    See Also
    ========

    factorial, factorial2, RisingFactorial

    References
    ==========

    .. [1] https://mathworld.wolfram.com/FallingFactorial.html
    .. [2] Peter Paule, "Greatest Factorial Factorization and Symbolic
           Summation", Journal of Symbolic Computation, vol. 20, pp. 235-268,
           1995.

    """

    @classmethod
    def eval(cls, x, k):
        x = sympify(x)
        k = sympify(k)

        if x is S.NaN or k is S.NaN:
            return S.NaN
        elif k.is_integer and x == k:
            return factorial(x)
        elif k.is_Integer:
            if k.is_zero:
                return S.One
            else:
                if k.is_positive:
                    if x is S.Infinity:
                        return S.Infinity
                    elif x is S.NegativeInfinity:
                        if k.is_odd:
                            return S.NegativeInfinity
                        else:
                            return S.Infinity
                    else:
                        if isinstance(x, Poly):
                            gens = x.gens
                            if len(gens)!= 1:
                                raise ValueError("ff only defined for "
                                            "polynomials on one generator")
                            else:
                                return reduce(lambda r, i:
                                              r*(x.shift(-i)),
                                              range(int(k)), 1)
                        else:
                            return reduce(lambda r, i: r*(x - i),
                                          range(int(k)), 1)
                else:
                    if x is S.Infinity:
                        return S.Infinity
                    elif x is S.NegativeInfinity:
                        return S.Infinity
                    else:
                        if isinstance(x, Poly):
                            gens = x.gens
                            if len(gens)!= 1:
                                raise ValueError("rf only defined for "
                                            "polynomials on one generator")
                            else:
                                return 1/reduce(lambda r, i:
                                                r*(x.shift(i)),
                                                range(1, abs(int(k)) + 1), 1)
                        else:
                            return 1/reduce(lambda r, i: r*(x + i),
                                            range(1, abs(int(k)) + 1), 1)

    def _eval_rewrite_as_gamma(self, x, k, piecewise=True, **kwargs):
        from sympy.functions.elementary.piecewise import Piecewise
        from sympy.functions.special.gamma_functions import gamma
        if not piecewise:
            if (x < 0) == True:
                return S.NegativeOne**k*gamma(k - x) / gamma(-x)
            return gamma(x + 1) / gamma(x - k + 1)
        return Piecewise(
            (gamma(x + 1) / gamma(x - k + 1), x >= 0),
            (S.NegativeOne**k*gamma(k - x) / gamma(-x), True))

    def _eval_rewrite_as_RisingFactorial(self, x, k, **kwargs):
        return rf(x - k + 1, k)

    def _eval_rewrite_as_binomial(self, x, k, **kwargs):
        if k.is_integer:
            return factorial(k) * binomial(x, k)

    def _eval_rewrite_as_factorial(self, x, k, **kwargs):
        from sympy.functions.elementary.piecewise import Piecewise
        if x.is_integer and k.is_integer:
            return Piecewise(
                (factorial(x)/factorial(-k + x), x >= 0),
                (S.NegativeOne**k*factorial(k - x - 1)/factorial(-x - 1), True))

    def _eval_rewrite_as_tractable(self, x, k, limitvar=None, **kwargs):
        from sympy.functions.special.gamma_functions import gamma
        if limitvar:
            k_lim = k.subs(limitvar, S.Infinity)
            if k_lim is S.Infinity:
                return (S.NegativeOne**k*gamma(k - x).rewrite('tractable', deep=True) / gamma(-x))
            elif k_lim is S.NegativeInfinity:
                return (gamma(x + 1) / gamma(x - k + 1).rewrite('tractable', deep=True))
        return self.rewrite(gamma).rewrite('tractable', deep=True)

    def _eval_is_integer(self):
        return fuzzy_and((self.args[0].is_integer, self.args[1].is_integer,
                          self.args[1].is_nonnegative))


rf = RisingFactorial
ff = FallingFactorial

###############################################################################
########################### BINOMIAL COEFFICIENTS #############################
###############################################################################


class binomial(CombinatorialFunction):
    r"""Implementation of the binomial coefficient. It can be defined
    in two ways depending on its desired interpretation:

    .. math:: \binom{n}{k} = \frac{n!}{k!(n-k)!}\ \text{or}\
                \binom{n}{k} = \frac{(n)_k}{k!}

    First, in a strict combinatorial sense it defines the
    number of ways we can choose `k` elements from a set of
    `n` elements. In this case both arguments are nonnegative
    integers and binomial is computed using an efficient
    algorithm based on prime factorization.

    The other definition is generalization for arbitrary `n`,
    however `k` must also be nonnegative. This case is very
    useful when evaluating summations.

    For the sake of convenience, for negative integer `k` this function
    will return zero no matter the other argument.

    To expand the binomial when `n` is a symbol, use either
    ``expand_func()`` or ``expand(func=True)``. The former will keep
    the polynomial in factored form while the latter will expand the
    polynomial itself. See examples for details.

    Examples
    ========

    >>> from sympy import Symbol, Rational, binomial, expand_func
    >>> n = Symbol('n', integer=True, positive=True)

    >>> binomial(15, 8)
    6435

    >>> binomial(n, -1)
    0

    Rows of Pascal's triangle can be generated with the binomial function:

    >>> for N in range(8):
    ...     print([binomial(N, i) for i in range(N + 1)])
    ...
    [1]
    [1, 1]
    [1, 2, 1]
    [1, 3, 3, 1]
    [1, 4, 6, 4, 1]
    [1, 5, 10, 10, 5, 1]
    [1, 6, 15, 20, 15, 6, 1]
    [1, 7, 21, 35, 35, 21, 7, 1]

    As can a given diagonal, e.g. the 4th diagonal:

    >>> N = -4
    >>> [binomial(N, i) for i in range(1 - N)]
    [1, -4, 10, -20, 35]

    >>> binomial(Rational(5, 4), 3)
    -5/128
    >>> binomial(Rational(-5, 4), 3)
    -195/128

    >>> binomial(n, 3)
    binomial(n, 3)

    >>> binomial(n, 3).expand(func=True)
    n**3/6 - n**2/2 + n/3

    >>> expand_func(binomial(n, 3))
    n*(n - 2)*(n - 1)/6

    In many cases, we can also compute binomial coefficients modulo a
    prime p quickly using Lucas' Theorem [2]_, though we need to include
    `evaluate=False` to postpone evaluation:

    >>> from sympy import Mod
    >>> Mod(binomial(156675, 4433, evaluate=False), 10**5 + 3)
    28625

    Using a generalisation of Lucas's Theorem given by Granville [3]_,
    we can extend this to arbitrary n:

    >>> Mod(binomial(10**18, 10**12, evaluate=False), (10**5 + 3)**2)
    3744312326

    References
    ==========

    .. [1] https://www.johndcook.com/blog/binomial_coefficients/
    .. [2] https://en.wikipedia.org/wiki/Lucas%27s_theorem
    .. [3] Binomial coefficients modulo prime powers, Andrew Granville,
        Available: https://web.archive.org/web/20170202003812/http://www.dms.umontreal.ca/~andrew/PDF/BinCoeff.pdf
    """

    def fdiff(self, argindex=1):
        from sympy.functions.special.gamma_functions import polygamma
        if argindex == 1:
            # https://functions.wolfram.com/GammaBetaErf/Binomial/20/01/01/
            n, k = self.args
            return binomial(n, k)*(polygamma(0, n + 1) - \
                polygamma(0, n - k + 1))
        elif argindex == 2:
            # https://functions.wolfram.com/GammaBetaErf/Binomial/20/01/02/
            n, k = self.args
            return binomial(n, k)*(polygamma(0, n - k + 1) - \
                polygamma(0, k + 1))
        else:
            raise ArgumentIndexError(self, argindex)

    @classmethod
    def _eval(self, n, k):
        # n.is_Number and k.is_Integer and k != 1 and n != k

        if k.is_Integer:
            if n.is_Integer and n >= 0:
                n, k = int(n), int(k)

                if k > n:
                    return S.Zero
                elif k > n // 2:
                    k = n - k

                if HAS_GMPY:
                    return Integer(gmpy.bincoef(n, k))

                d, result = n - k, 1
                for i in range(1, k + 1):
                    d += 1
                    result = result * d // i
                return Integer(result)
            else:
                d, result = n - k, 1
                for i in range(1, k + 1):
                    d += 1
                    result *= d
                return result / _factorial(k)

    @classmethod
    def eval(cls, n, k):
        n, k = map(sympify, (n, k))
        d = n - k
        n_nonneg, n_isint = n.is_nonnegative, n.is_integer
        if k.is_zero or ((n_nonneg or n_isint is False)
                and d.is_zero):
            return S.One
        if (k - 1).is_zero or ((n_nonneg or n_isint is False)
                and (d - 1).is_zero):
            return n
        if k.is_integer:
            if k.is_negative or (n_nonneg and n_isint and d.is_negative):
                return S.Zero
            elif n.is_number:
                res = cls._eval(n, k)
                return res.expand(basic=True) if res else res
        elif n_nonneg is False and n_isint:
            # a special case when binomial evaluates to complex infinity
            return S.ComplexInfinity
        elif k.is_number:
            from sympy.functions.special.gamma_functions import gamma
            return gamma(n + 1)/(gamma(k + 1)*gamma(n - k + 1))

    def _eval_Mod(self, q):
        n, k = self.args

        if any(x.is_integer is False for x in (n, k, q)):
            raise ValueError("Integers expected for binomial Mod")

        if all(x.is_Integer for x in (n, k, q)):
            n, k = map(int, (n, k))
            aq, res = abs(q), 1

            # handle negative integers k or n
            if k < 0:
                return S.Zero
            if n < 0:
                n = -n + k - 1
                res = -1 if k%2 else 1

            # non negative integers k and n
            if k > n:
                return S.Zero

            isprime = aq.is_prime
            aq = int(aq)
            if isprime:
                if aq < n:
                    # use Lucas Theorem
                    N, K = n, k
                    while N or K:
                        res = res*binomial(N % aq, K % aq) % aq
                        N, K = N // aq, K // aq

                else:
                    # use Factorial Modulo
                    d = n - k
                    if k > d:
                        k, d = d, k
                    kf = 1
                    for i in range(2, k + 1):
                        kf = kf*i % aq
                    df = kf
                    for i in range(k + 1, d + 1):
                        df = df*i % aq
                    res *= df
                    for i in range(d + 1, n + 1):
                        res = res*i % aq

                    res *= pow(kf*df % aq, aq - 2, aq)
                    res %= aq

            elif _sqrt(q) < k and q != 1:
                res = binomial_mod(n, k, q)

            else:
                # Binomial Factorization is performed by calculating the
                # exponents of primes <= n in `n! /(k! (n - k)!)`,
                # for non-negative integers n and k. As the exponent of
                # prime in n! is e_p(n) = [n/p] + [n/p**2] + ...
                # the exponent of prime in binomial(n, k) would be
                # e_p(n) - e_p(k) - e_p(n - k)
                M = int(_sqrt(n))
                for prime in sieve.primerange(2, n + 1):
                    if prime > n - k:
                        res = res*prime % aq
                    elif prime > n // 2:
                        continue
                    elif prime > M:
                        if n % prime < k % prime:
                            res = res*prime % aq
                    else:
                        N, K = n, k
                        exp = a = 0

                        while N > 0:
                            a = int((N % prime) < (K % prime + a))
                            N, K = N // prime, K // prime
                            exp += a

                        if exp > 0:
                            res *= pow(prime, exp, aq)
                            res %= aq

            return S(res % q)

    def _eval_expand_func(self, **hints):
        """
        Function to expand binomial(n, k) when m is positive integer
        Also,
        n is self.args[0] and k is self.args[1] while using binomial(n, k)
        """
        n = self.args[0]
        if n.is_Number:
            return binomial(*self.args)

        k = self.args[1]
        if (n-k).is_Integer:
            k = n - k

        if k.is_Integer:
            if k.is_zero:
                return S.One
            elif k.is_negative:
                return S.Zero
            else:
                n, result = self.args[0], 1
                for i in range(1, k + 1):
                    result *= n - k + i
                return result / _factorial(k)
        else:
            return binomial(*self.args)

    def _eval_rewrite_as_factorial(self, n, k, **kwargs):
        return factorial(n)/(factorial(k)*factorial(n - k))

    def _eval_rewrite_as_gamma(self, n, k, piecewise=True, **kwargs):
        from sympy.functions.special.gamma_functions import gamma
        return gamma(n + 1)/(gamma(k + 1)*gamma(n - k + 1))

    def _eval_rewrite_as_tractable(self, n, k, limitvar=None, **kwargs):
        return self._eval_rewrite_as_gamma(n, k).rewrite('tractable')

    def _eval_rewrite_as_FallingFactorial(self, n, k, **kwargs):
        if k.is_integer:
            return ff(n, k) / factorial(k)

    def _eval_is_integer(self):
        n, k = self.args
        if n.is_integer and k.is_integer:
            return True
        elif k.is_integer is False:
            return False

    def _eval_is_nonnegative(self):
        n, k = self.args
        if n.is_integer and k.is_integer:
            if n.is_nonnegative or k.is_negative or k.is_even:
                return True
            elif k.is_even is False:
                return  False

    def _eval_as_leading_term(self, x, logx=None, cdir=0):
        from sympy.functions.special.gamma_functions import gamma
        return self.rewrite(gamma)._eval_as_leading_term(x, logx=logx, cdir=cdir)
