Generating function of f ( n ) = C n − ∑ k = 1...
Generating function of .
Answer & Explanation
There are multiple types of generating functions (ordinary, exponential etc.). Your question is non-specific, so we'll go for the exponential generating function since the second part of the recurrence function looks like a binomial convolution.
So let's compute
Then we have
where (1) uses the generating function for Catalan numbers, and in (2) we use the formula for the product of two series (Cauchy product). Here and represent the modified Bessel functions of the first kind.
Using , we obtain from (3):
If you were after the ordinary generating function
you can obtain it, if it converges, from the exponential one here, including