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FindGeneratingFunction[{a0,a1,…},x]

attempts to find a simple generating function in x whose n^(th) series coefficient is an.

FindGeneratingFunction[{{n0,a0},{n1,a1},…},x]

attempts to find a simple generating function whose ni^(th) series coefficient is ai.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Generalizations & Extensions  
Properties & Relations  
See Also
Related Guides
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FindGeneratingFunction

FindGeneratingFunction[{a0,a1,…},x]

attempts to find a simple generating function in x whose n^(th) series coefficient is an.

FindGeneratingFunction[{{n0,a0},{n1,a1},…},x]

attempts to find a simple generating function whose ni^(th) series coefficient is ai.

Details and Options

  • The sequence elements an can be either exact numbers or symbolic expressions.
  • FindGeneratingFunction finds results in terms of a wide range of integer functions, as well as implicit solutions to difference equations represented by DifferenceRoot.
  • If FindGeneratingFunction cannot find a simple generating function that yields the specified sequence, it returns unevaluated.
  • FindGeneratingFunction has the following options:
  • FunctionSpaceAutomaticwhere to look for candidate simple generating functions
    MethodAutomaticmethod to use
    TimeConstraint10how many seconds to search a particular function space or perform a transformation
    ValidationLengthAutomaticsequence length used to validate a candidate generating function found
  • FindGeneratingFunction[list,x] by default uses earlier elements in list to find candidate simple generating functions, then validates the generating functions by looking at later elements.
  • FindGeneratingFunction[list,x] only returns functions that correctly reproduce all elements of list.

Examples

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Basic Examples  (2)

Find a generating function for a sequence:

Wolfram Language code: FindGeneratingFunction[{1, 1, 2, 3, 5, 8, 13}, x]
Wolfram Language code: Series[%, {x, 0, 7}]

A periodic sequence:

Wolfram Language code: FindGeneratingFunction[{0, 1, 2, 0, 1, 2, 0}, x]//Together
Wolfram Language code: Series[%, {x, 0, 7}]

Scope  (2)

Rational functions:

Wolfram Language code: FindGeneratingFunction[{1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 11, 12}, x]

Hypergeometric functions:

Wolfram Language code: FindGeneratingFunction[{1, 3 / 2, 9 / 5, 2, 15 / 7, 9 / 4, 7 / 3, 12 / 5}, x]

Generalizations & Extensions  (1)

FindGeneratingFunction works on arbitrary exact numbers or symbolic expressions:

Wolfram Language code: FindGeneratingFunction[{1 + E, Sqrt[2] + E, 2 + E, 2Sqrt[2] + E, 4 + E}, n]//Apart
Wolfram Language code: FindGeneratingFunction[{1 + a, 1 + a ^ 2, 1 + a ^ 3, 1 + a ^ 4, 1 + a ^ 5}, n]

Properties & Relations  (1)

Use FindSequenceFunction to find a generating function of a sequence:

Wolfram Language code: FindSequenceFunction[{3, 7, 13, 21, 31, 43, 57}, n]
Wolfram Language code: GeneratingFunction[% /. n -> n + 1, n, x]

Verify:

Wolfram Language code: FindGeneratingFunction[{3, 7, 13, 21, 31, 43, 57}, x]//Together
Wolfram Research (2008), FindGeneratingFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/FindGeneratingFunction.html.

Text

Wolfram Research (2008), FindGeneratingFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/FindGeneratingFunction.html.

CMS

Wolfram Language. 2008. "FindGeneratingFunction." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FindGeneratingFunction.html.

APA

Wolfram Language. (2008). FindGeneratingFunction. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FindGeneratingFunction.html

BibTeX

@misc{reference.wolfram_2026_findgeneratingfunction, author="Wolfram Research", title="{FindGeneratingFunction}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/FindGeneratingFunction.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_findgeneratingfunction, organization={Wolfram Research}, title={FindGeneratingFunction}, year={2008}, url={https://reference.wolfram.com/language/ref/FindGeneratingFunction.html}, note=[Accessed: 12-June-2026]}