Function from Taylor series?

Nathan Urban · science forum beginner Joined: 16 Jul 2006 Posts: 2

I have a series that looks like this:

g(x) = a_0 f(0) + a_1 f''(0) x^2 + a_2 f^(4)(0) x^4 + ...

where f(x) is an arbitrary even function and a_i are coefficients.

This looks like a Taylor series expansion of f(x) about x=0, except
that the a_i are not the usual 1/n! for Taylor series.

I would like to know if, given the a_i, one can find a closed-form
expression for g(x), expressed somehow in terms of the unknown
function f(x), for which the above series is the Taylor expansion?

For concreteness, the actual coefficients I'm looking at are a_i = 1/S_i,
where S_i is the ith term in Series A110468 from the Encyclopedia
of Integer Sequences, "Convolution of (-1)^n*n! and n!".

Robert B. Israel · science forum Guru Joined: 24 Mar 2005 Posts: 2151

In article <e9mjtd$5pc$1@dizzy.math.ohio-state.edu>,
Nathan Urban <nurban@psu.edu> wrote:

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