code: def test_genocchi():
      genocchis = [0, -1, -1, 0, 1, 0, -3, 0, 17]
      for n, g in enumerate(genocchis):
          assert genocchi(n) == g
  
      m = Symbol("m", integer=True)
      n = Symbol("n", integer=True, positive=True)
      assert unchanged(genocchi, m)
      assert genocchi(2 * n + 1) == 0
      gn = 2 * (1 - 2**n) * bernoulli(n)
      assert genocchi(n).rewrite(bernoulli).factor() == gn.factor()
      gnx = 2 * (bernoulli(n, x) - 2**n * bernoulli(n, (x + 1) / 2))
      assert genocchi(n, x).rewrite(bernoulli).factor() == gnx.factor()
      assert genocchi(2 * n).is_odd
      assert genocchi(2 * n).is_even is False
      assert genocchi(2 * n + 1).is_even
      assert genocchi(n).is_integer
      assert genocchi(4 * n).is_positive
      # these are the only 2 prime Genocchi numbers
      assert genocchi(6, evaluate=False).is_prime == S(-3).is_prime
      assert genocchi(8, evaluate=False).is_prime
      assert genocchi(4 * n + 2).is_negative
      assert genocchi(4 * n + 1).is_negative is False
      assert genocchi(4 * n - 2).is_negative
  
      g0 = genocchi(0, evaluate=False)
      assert g0.is_positive is False
      assert g0.is_negative is False
      assert g0.is_even is True
      assert g0.is_odd is False
  
      assert genocchi(0, x) == 0
      assert genocchi(1, x) == -1
      assert genocchi(2, x) == 1 - 2 * x
      assert genocchi(3, x) == 3 * x - 3 * x**2
      assert genocchi(4, x) == -1 + 6 * x**2 - 4 * x**3
      y = Symbol("y")
      assert (
          genocchi(5, (x + y) ** 100)
          == -5 * (x + y) ** 400
          + 10 * (x + y) ** 300
          - 5 * (x + y) ** 100
      )
  
      assert str(genocchi(5.0, 4.0).evalf(n=10)) == "-660.0000000"
      assert str(genocchi(Rational(5, 4)).evalf(n=10)) == "-1.104286457"
      assert str(genocchi(-2).evalf(n=10)) == "3.606170709"
      assert str(genocchi(1.3, 3.7).evalf(n=10)) == "-1.847375373"
      assert (
          str(genocchi(I, 1.0).evalf(n=10))
          == "-0.3161917278 - 1.45311955*I"
      )
  
      n = Symbol("n")
      assert genocchi(n, x).rewrite(
          dirichlet_eta
      ) == -2 * n * dirichlet_eta(1 - n, x)
  
imports: import string
  from sympy.concrete.products import Product
  from sympy.concrete.summations import Sum
  from sympy.core.function import diff, expand_func
  from sympy.core import EulerGamma, TribonacciConstant
  from sympy.core.numbers import Float, I, Rational, oo, pi
  from sympy.core.singleton import S
  from sympy.core.symbol import Dummy, Symbol, symbols
  from sympy.functions.combinatorial.numbers import carmichael
  from sympy.functions.elementary.complexes import im, re
  from sympy.functions.elementary.integers import floor
  from sympy.polys.polytools import cancel
  from sympy.series.limits import limit, Limit
  from sympy.series.order import O
  from sympy.functions import bernoulli, harmonic, bell, fibonacci, tribonacci, lucas, euler, catalan, genocchi, andre, partition, mobius, motzkin, binomial, gamma, sqrt, cbrt, hyper, log, digamma, trigamma, polygamma, factorial, sin, cos, cot, polylog, zeta, dirichlet_eta
  from sympy.functions.combinatorial.numbers import _nT
  from sympy.core.expr import unchanged
  from sympy.core.numbers import GoldenRatio, Integer
  from sympy.testing.pytest import raises, nocache_fail, warns_deprecated_sympy
  from sympy.abc import x
label_desc: Use image node_id 17 for calling a global function with example usage: test_genocchi() without return types