code: def test_gosper_sum_AeqB_part1():
      f1a = n**4
      f1b = n**3 * 2**n
      f1c = 1 / (n**2 + sqrt(5) * n - 1)
      f1d = n**4 * 4**n / binomial(2 * n, n)
      f1e = factorial(3 * n) / (
          factorial(n) * factorial(n + 1) * factorial(n + 2) * 27**n
      )
      f1f = binomial(2 * n, n) ** 2 / ((n + 1) * 4 ** (2 * n))
      f1g = (
          (4 * n - 1)
          * binomial(2 * n, n) ** 2
          / ((2 * n - 1) ** 2 * 4 ** (2 * n))
      )
      f1h = n * factorial(n - S.Half) ** 2 / factorial(n + 1) ** 2
  
      g1a = m * (m + 1) * (2 * m + 1) * (3 * m**2 + 3 * m - 1) / 30
      g1b = 26 + 2 ** (m + 1) * (m**3 - 3 * m**2 + 9 * m - 13)
      g1c = (
          (m + 1)
          * (
              m * (m**2 - 7 * m + 3) * sqrt(5)
              - (3 * m**3 - 7 * m**2 + 19 * m - 6)
          )
          / (2 * m**3 * sqrt(5) + m**4 + 5 * m**2 - 1)
          / 6
      )
      g1d = Rational(-2, 231) + 2 * 4**m * (m + 1) * (
          63 * m**4 + 112 * m**3 + 18 * m**2 - 22 * m + 3
      ) / (693 * binomial(2 * m, m))
      g1e = Rational(-9, 2) + (81 * m**2 + 261 * m + 200) * factorial(
          3 * m + 2
      ) / (
          40
          * 27**m
          * factorial(m)
          * factorial(m + 1)
          * factorial(m + 2)
      )
      g1f = (
          (2 * m + 1) ** 2
          * binomial(2 * m, m) ** 2
          / (4 ** (2 * m) * (m + 1))
      )
      g1g = -binomial(2 * m, m) ** 2 / 4 ** (2 * m)
      g1h = (
          4 * pi
          - (2 * m + 1) ** 2
          * (3 * m + 4)
          * factorial(m - S.Half) ** 2
          / factorial(m + 1) ** 2
      )
  
      g = gosper_sum(f1a, (n, 0, m))
      assert g is not None and simplify(g - g1a) == 0
      g = gosper_sum(f1b, (n, 0, m))
      assert g is not None and simplify(g - g1b) == 0
      g = gosper_sum(f1c, (n, 0, m))
      assert g is not None and simplify(g - g1c) == 0
      g = gosper_sum(f1d, (n, 0, m))
      assert g is not None and simplify(g - g1d) == 0
      g = gosper_sum(f1e, (n, 0, m))
      assert g is not None and simplify(g - g1e) == 0
      g = gosper_sum(f1f, (n, 0, m))
      assert g is not None and simplify(g - g1f) == 0
      g = gosper_sum(f1g, (n, 0, m))
      assert g is not None and simplify(g - g1g) == 0
      g = gosper_sum(f1h, (n, 0, m))
      # need to call rewrite(gamma) here because we have terms involving
      # factorial(1/2)
      assert g is not None and simplify(g - g1h).rewrite(gamma) == 0
  
imports: from sympy.core.numbers import Rational, pi
  from sympy.core.singleton import S
  from sympy.core.symbol import Symbol
  from sympy.functions.combinatorial.factorials import binomial, factorial
  from sympy.functions.elementary.exponential import exp, log
  from sympy.functions.elementary.miscellaneous import sqrt
  from sympy.functions.special.gamma_functions import gamma
  from sympy.polys.polytools import Poly
  from sympy.simplify.simplify import simplify
  from sympy.concrete.gosper import gosper_normal, gosper_sum, gosper_term
  from sympy.abc import a, b, j, k, m, n, r, x
label_desc: Use image node_id 8 for calling a global function with example usage: test_gosper_sum_AeqB_part1() without return types