I have a series $\{g_n\}$ whose values are hard to compute, but I calculated a generating function for it (I know the square root is unconventional, but it results in a nice exponential function):
$$ \sum_n \frac{x^n}{\sqrt{n!}}g_n = e^{f(x)} $$
Now, I am interested in the inner product of a sequence $h_n$ with $g_n$:
$$ Y = \sum_n g_nh_n $$
However, I don't have a generating function for $h_n$ because it's a sequence of numerical values.
Is there a way of combining the generating function $e^{f(x)}$ with $h_n$ to compute $Y$?
I have found that two generating functions could be combined via a Hadamard product (so I would need something like $F\odot G(1)$, barring the $\sqrt{n!}$ factor), but I'm not sure how that can help.