code: def guess_generating_function(
      v, X=Symbol("x"), types=["all"], maxsqrtn=2
  ):
      """
      Tries to "guess" a generating function for a sequence of rational numbers v.
      Only a few patterns are implemented yet.
  
      Explanation
      ===========
  
      The function returns a dictionary where keys are the name of a given type of
      generating function. Six types are currently implemented:
  
           type  |  formal definition
          -------+----------------------------------------------------------------
          ogf    | f(x) = Sum(            a_k * x^k       ,  k: 0..infinity )
          egf    | f(x) = Sum(            a_k * x^k / k!  ,  k: 0..infinity )
          lgf    | f(x) = Sum( (-1)^(k+1) a_k * x^k / k   ,  k: 1..infinity )
                 |        (with initial index being hold as 1 rather than 0)
          hlgf   | f(x) = Sum(            a_k * x^k / k   ,  k: 1..infinity )
                 |        (with initial index being hold as 1 rather than 0)
          lgdogf | f(x) = derivate( log(Sum( a_k * x^k, k: 0..infinity )), x)
          lgdegf | f(x) = derivate( log(Sum( a_k * x^k / k!, k: 0..infinity )), x)
  
      In order to spare time, the user can select only some types of generating
      functions (default being ['all']). While forgetting to use a list in the
      case of a single type may seem to work most of the time as in: types='ogf'
      this (convenient) syntax may lead to unexpected extra results in some cases.
  
      Discarding a type when calling the function does not mean that the type will
      not be present in the returned dictionary; it only means that no extra
      computation will be performed for that type, but the function may still add
      it in the result when it can be easily converted from another type.
  
      Two generating functions (lgdogf and lgdegf) are not even computed if the
      initial term of the sequence is 0; it may be useful in that case to try
      again after having removed the leading zeros.
  
      Examples
      ========
  
      >>> from sympy.concrete.guess import guess_generating_function as ggf
      >>> ggf([k+1 for k in range(12)], types=['ogf', 'lgf', 'hlgf'])
      {'hlgf': 1/(1 - x), 'lgf': 1/(x + 1), 'ogf': 1/(x**2 - 2*x + 1)}
  
      >>> from sympy import sympify
      >>> l = sympify("[3/2, 11/2, 0, -121/2, -363/2, 121]")
      >>> ggf(l)
      {'ogf': (x + 3/2)/(11*x**2 - 3*x + 1)}
  
      >>> from sympy import fibonacci
      >>> ggf([fibonacci(k) for k in range(5, 15)], types=['ogf'])
      {'ogf': (3*x + 5)/(-x**2 - x + 1)}
  
      >>> from sympy import factorial
      >>> ggf([factorial(k) for k in range(12)], types=['ogf', 'egf', 'lgf'])
      {'egf': 1/(1 - x)}
  
      >>> ggf([k+1 for k in range(12)], types=['egf'])
      {'egf': (x + 1)*exp(x), 'lgdegf': (x + 2)/(x + 1)}
  
      N-th root of a rational function can also be detected (below is an example
      coming from the sequence A108626 from https://oeis.org).
      The greatest n-th root to be tested is specified as maxsqrtn (default 2).
  
      >>> ggf([1, 2, 5, 14, 41, 124, 383, 1200, 3799, 12122, 38919])['ogf']
      sqrt(1/(x**4 + 2*x**2 - 4*x + 1))
  
      References
      ==========
  
      .. [1] "Concrete Mathematics", R.L. Graham, D.E. Knuth, O. Patashnik
      .. [2] https://oeis.org/wiki/Generating_functions
  
      """
      # List of all types of all g.f. known by the algorithm
      if "all" in types:
          types = ("ogf", "egf", "lgf", "hlgf", "lgdogf", "lgdegf")
  
      result = {}
  
      # Ordinary Generating Function (ogf)
      if "ogf" in types:
          # Perform some convolutions of the sequence with itself
          t = [1] + [0] * (len(v) - 1)
          for d in range(max(1, maxsqrtn)):
              t = [
                  sum(t[n - i] * v[i] for i in range(n + 1))
                  for n in range(len(v))
              ]
              g = guess_generating_function_rational(t, X=X)
              if g:
                  result["ogf"] = g ** Rational(1, d + 1)
                  break
  
      # Exponential Generating Function (egf)
      if "egf" in types:
          # Transform sequence (division by factorial)
          w, f = [], S.One
          for i, k in enumerate(v):
              f *= i if i else 1
              w.append(k / f)
          # Perform some convolutions of the sequence with itself
          t = [1] + [0] * (len(w) - 1)
          for d in range(max(1, maxsqrtn)):
              t = [
                  sum(t[n - i] * w[i] for i in range(n + 1))
                  for n in range(len(w))
              ]
              g = guess_generating_function_rational(t, X=X)
              if g:
                  result["egf"] = g ** Rational(1, d + 1)
                  break
  
      # Logarithmic Generating Function (lgf)
      if "lgf" in types:
          # Transform sequence (multiplication by (-1)^(n+1) / n)
          w, f = [], S.NegativeOne
          for i, k in enumerate(v):
              f = -f
              w.append(f * k / Integer(i + 1))
          # Perform some convolutions of the sequence with itself
          t = [1] + [0] * (len(w) - 1)
          for d in range(max(1, maxsqrtn)):
              t = [
                  sum(t[n - i] * w[i] for i in range(n + 1))
                  for n in range(len(w))
              ]
              g = guess_generating_function_rational(t, X=X)
              if g:
                  result["lgf"] = g ** Rational(1, d + 1)
                  break
  
      # Hyperbolic logarithmic Generating Function (hlgf)
      if "hlgf" in types:
          # Transform sequence (division by n+1)
          w = []
          for i, k in enumerate(v):
              w.append(k / Integer(i + 1))
          # Perform some convolutions of the sequence with itself
          t = [1] + [0] * (len(w) - 1)
          for d in range(max(1, maxsqrtn)):
              t = [
                  sum(t[n - i] * w[i] for i in range(n + 1))
                  for n in range(len(w))
              ]
              g = guess_generating_function_rational(t, X=X)
              if g:
                  result["hlgf"] = g ** Rational(1, d + 1)
                  break
  
      # Logarithmic derivative of ordinary generating Function (lgdogf)
      if v[0] != 0 and (
          "lgdogf" in types or ("ogf" in types and "ogf" not in result)
      ):
          # Transform sequence by computing f'(x)/f(x)
          # because log(f(x)) = integrate( f'(x)/f(x) )
          a, w = sympify(v[0]), []
          for n in range(len(v) - 1):
              w.append(
                  (
                      v[n + 1] * (n + 1)
                      - sum(w[-i - 1] * v[i + 1] for i in range(n))
                  )
                  / a
              )
          # Perform some convolutions of the sequence with itself
          t = [1] + [0] * (len(w) - 1)
          for d in range(max(1, maxsqrtn)):
              t = [
                  sum(t[n - i] * w[i] for i in range(n + 1))
                  for n in range(len(w))
              ]
              g = guess_generating_function_rational(t, X=X)
              if g:
                  result["lgdogf"] = g ** Rational(1, d + 1)
                  if "ogf" not in result:
                      result["ogf"] = exp(
                          integrate(result["lgdogf"], X)
                      )
                  break
  
      # Logarithmic derivative of exponential generating Function (lgdegf)
      if v[0] != 0 and (
          "lgdegf" in types or ("egf" in types and "egf" not in result)
      ):
          # Transform sequence / step 1 (division by factorial)
          z, f = [], S.One
          for i, k in enumerate(v):
              f *= i if i else 1
              z.append(k / f)
          # Transform sequence / step 2 by computing f'(x)/f(x)
          # because log(f(x)) = integrate( f'(x)/f(x) )
          a, w = z[0], []
          for n in range(len(z) - 1):
              w.append(
                  (
                      z[n + 1] * (n + 1)
                      - sum(w[-i - 1] * z[i + 1] for i in range(n))
                  )
                  / a
              )
          # Perform some convolutions of the sequence with itself
          t = [1] + [0] * (len(w) - 1)
          for d in range(max(1, maxsqrtn)):
              t = [
                  sum(t[n - i] * w[i] for i in range(n + 1))
                  for n in range(len(w))
              ]
              g = guess_generating_function_rational(t, X=X)
              if g:
                  result["lgdegf"] = g ** Rational(1, d + 1)
                  if "egf" not in result:
                      result["egf"] = exp(
                          integrate(result["lgdegf"], X)
                      )
                  break
  
      return result
  
file_path: sympy/sympy/concrete/guess.py
imports: from sympy.concrete.products import Product, product
  from sympy.core import Function, S
  from sympy.core.add import Add
  from sympy.core.numbers import Integer, Rational
  from sympy.core.symbol import Symbol, symbols
  from sympy.core.sympify import sympify
  from sympy.functions.elementary.exponential import exp
  from sympy.functions.elementary.integers import floor
  from sympy.integrals.integrals import integrate
  from sympy.polys.polyfuncs import rational_interpolate
  from sympy.polys.polytools import lcm
  from sympy.simplify.radsimp import denom
  from sympy.utilities import public
label_desc: Use image node_id 5 for calling a global function with example usage: guess_generating_function(v, X, types, maxsqrtn) and returns: result