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GeneratingFunction

Compute generating function of a sequence.

$ wo 'GeneratingFunction[a, n, x]'
a/(1 - x)

The Fibonacci and Lucas sequences share the denominator 1 - x - x^2.

$ wo 'GeneratingFunction[Fibonacci[n], n, x]'
-(x/(-1 + x + x^2))

$ wo 'GeneratingFunction[LucasL[n], n, x]'
(-2 + x)/(-1 + x + x^2)

The Catalan numbers and the reciprocal-shift sequence.

$ wo 'GeneratingFunction[CatalanNumber[n], n, x]'
2/(1 + Sqrt[1 - 4*x])

$ wo 'GeneratingFunction[1/(n + 2), n, x]'
(-1 - Log[1 - x]/x)/x