code: def test_factorial2():
      n = Symbol("n", integer=True)
  
      assert factorial2(-1) == 1
      assert factorial2(0) == 1
      assert factorial2(7) == 105
      assert factorial2(8) == 384
  
      # The following is exhaustive
      tt = Symbol("tt", integer=True, nonnegative=True)
      tte = Symbol("tte", even=True, nonnegative=True)
      tpe = Symbol("tpe", even=True, positive=True)
      tto = Symbol("tto", odd=True, nonnegative=True)
      tf = Symbol("tf", integer=True, nonnegative=False)
      tfe = Symbol("tfe", even=True, nonnegative=False)
      tfo = Symbol("tfo", odd=True, nonnegative=False)
      ft = Symbol("ft", integer=False, nonnegative=True)
      ff = Symbol("ff", integer=False, nonnegative=False)
      fn = Symbol("fn", integer=False)
      nt = Symbol("nt", nonnegative=True)
      nf = Symbol("nf", nonnegative=False)
      nn = Symbol("nn")
      z = Symbol("z", zero=True)
      # Solves and Fixes Issue #10388 - This is the updated test for the same solved issue
      raises(ValueError, lambda: factorial2(oo))
      raises(ValueError, lambda: factorial2(Rational(5, 2)))
      raises(ValueError, lambda: factorial2(-4))
      assert factorial2(n).is_integer is None
      assert factorial2(tt - 1).is_integer
      assert factorial2(tte - 1).is_integer
      assert factorial2(tpe - 3).is_integer
      assert factorial2(tto - 4).is_integer
      assert factorial2(tto - 2).is_integer
      assert factorial2(tf).is_integer is None
      assert factorial2(tfe).is_integer is None
      assert factorial2(tfo).is_integer is None
      assert factorial2(ft).is_integer is None
      assert factorial2(ff).is_integer is None
      assert factorial2(fn).is_integer is None
      assert factorial2(nt).is_integer is None
      assert factorial2(nf).is_integer is None
      assert factorial2(nn).is_integer is None
  
      assert factorial2(n).is_positive is None
      assert factorial2(tt - 1).is_positive is True
      assert factorial2(tte - 1).is_positive is True
      assert factorial2(tpe - 3).is_positive is True
      assert factorial2(tpe - 1).is_positive is True
      assert factorial2(tto - 2).is_positive is True
      assert factorial2(tto - 1).is_positive is True
      assert factorial2(tf).is_positive is None
      assert factorial2(tfe).is_positive is None
      assert factorial2(tfo).is_positive is None
      assert factorial2(ft).is_positive is None
      assert factorial2(ff).is_positive is None
      assert factorial2(fn).is_positive is None
      assert factorial2(nt).is_positive is None
      assert factorial2(nf).is_positive is None
      assert factorial2(nn).is_positive is None
  
      assert factorial2(tt).is_even is None
      assert factorial2(tt).is_odd is None
      assert factorial2(tte).is_even is None
      assert factorial2(tte).is_odd is None
      assert factorial2(tte + 2).is_even is True
      assert factorial2(tpe).is_even is True
      assert factorial2(tpe).is_odd is False
      assert factorial2(tto).is_odd is True
      assert factorial2(tf).is_even is None
      assert factorial2(tf).is_odd is None
      assert factorial2(tfe).is_even is None
      assert factorial2(tfe).is_odd is None
      assert factorial2(tfo).is_even is False
      assert factorial2(tfo).is_odd is None
      assert factorial2(z).is_even is False
      assert factorial2(z).is_odd is True
  
imports: from sympy.concrete.products import Product
  from sympy.core.function import expand_func
  from sympy.core.mod import Mod
  from sympy.core.mul import Mul
  from sympy.core import EulerGamma
  from sympy.core.numbers import Float, I, Rational, nan, oo, pi, zoo
  from sympy.core.relational import Eq
  from sympy.core.singleton import S
  from sympy.core.symbol import Dummy, Symbol, symbols
  from sympy.functions.combinatorial.factorials import ff, rf, binomial, factorial, factorial2
  from sympy.functions.elementary.miscellaneous import sqrt
  from sympy.functions.elementary.piecewise import Piecewise
  from sympy.functions.special.gamma_functions import gamma, polygamma
  from sympy.polys.polytools import Poly
  from sympy.series.order import O
  from sympy.simplify.simplify import simplify
  from sympy.core.expr import unchanged
  from sympy.core.function import ArgumentIndexError
  from sympy.functions.combinatorial.factorials import subfactorial
  from sympy.functions.special.gamma_functions import uppergamma
  from sympy.testing.pytest import XFAIL, raises, slow
label_desc: Use image node_id 10 for calling a global function with example usage: test_factorial2() without return types