({{, (RefLink[FindGeneratingFunction, paclet:ref/FindGeneratingFunction])[{(Subscript[a, 0]),(Subscript[a, 1]),…},x] 
attempts to find a simple generating function in x whose n(^th) series coefficient is (Subscript[a, n]).}, {, (RefLink[FindGeneratingFunction, paclet:ref/FindGeneratingFunction])[{{(Subscript[n, 0]),(Subscript[a, 0])},{(Subscript[n, 1]),(Subscript[a, 1])},…},x]
attempts to find a simple generating function whose (Subscript[n, i])(^th) series coefficient is (Subscript[a, i]). }})

OpenersNotesSection(Spacer1)Details and Options

The sequence elements (Subscript[a, n]) can be either exact numbers or symbolic expressions.

(RefLink[FindGeneratingFunction, paclet:ref/FindGeneratingFunction]) finds results in terms of a wide range of integer functions, as well as implicit solutions to difference equations represented by (RefLink[DifferenceRoot, paclet:ref/DifferenceRoot]).

If (RefLink[FindGeneratingFunction, paclet:ref/FindGeneratingFunction]) cannot find a simple generating function that yields the specified sequence, it returns unevaluated.

(RefLink[FindGeneratingFunction, paclet:ref/FindGeneratingFunction]) has the following options:

(RefLink[FindGeneratingFunction, paclet:ref/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.

(RefLink[FindGeneratingFunction, paclet:ref/FindGeneratingFunction])[list,x] only returns functions that correctly reproduce all elements of list.

OpenersPrimaryExamplesSection(Spacer1)Examples  (6)

OpenersExampleSection0(Spacer1)Basic Examples  (2)

Find a generating function for a sequence:

FindGeneratingFunction[{1,1,2,3,5,8,13},x]

Series[%,{x,0,7}]

A periodic sequence:

FindGeneratingFunction[{0,1,2,0,1,2,0},x]//Together

Series[%,{x,0,7}]

OpenersExampleSection1(Spacer1)Scope  (2)

Rational functions:

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

Hypergeometric functions:

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

OpenersExampleSection2(Spacer1)Generalizations & Extensions  (1)

(RefLink[FindGeneratingFunction, paclet:ref/FindGeneratingFunction]) works on arbitrary exact numbers or symbolic expressions:

FindGeneratingFunction[{1+E,Sqrt[2]+E,2+E,2Sqrt[2]+E,4+E},n]//Apart

FindGeneratingFunction[{1+a,1+a^2,1+a^3,1+a^4,1+a^5},n]

OpenersExampleSection4(Spacer1)Properties & Relations  (1)

Use (RefLink[FindSequenceFunction, paclet:ref/FindSequenceFunction]) to find a generating function of a sequence:

FindSequenceFunction[{3,7,13,21,31,43,57},n]

GeneratingFunction[% /. n-> n+1, n, x]

Verify:

FindGeneratingFunction[{3,7,13,21,31,43,57},x]//Together

({{DocumentationFooterIconsRelatedFunction.png, ({{See Also}, {(RefLink[FindSequenceFunction, paclet:ref/FindSequenceFunction])  ▪  (RefLink[GeneratingFunction, paclet:ref/GeneratingFunction])  ▪  (RefLink[DifferentialRoot, paclet:ref/DifferentialRoot])  ▪  (RefLink[Series, paclet:ref/Series])  ▪  (RefLink[SeriesCoefficient, paclet:ref/SeriesCoefficient])  ▪  (RefLink[FindLinearRecurrence, paclet:ref/FindLinearRecurrence])}})}})

({{DocumentationFooterIconsRelatedGuide.png, ({{Related Guides}, {▪(RefLinkPlain[Integer Sequences, paclet:guide/IntegerSequences])}, {▪(RefLinkPlain[Discrete Calculus, paclet:guide/DiscreteCalculus])}})}})