I'm trying to find a closed form expression for the following sequence:
$$a_n=\sum_{i=1}^{n}\frac{(n-1-i+d)!}{(n-2i)!(i)!}=\sum_{i=1}^{\frac{n}{2}}\frac{(n-1-i+d)!}{(n-2i)!(i)!}$$
Where $n$ and $d$ are both positive integers.
An important note!: I'm actually not even interested in an exact solution. Any closed form that is a 'reasonably' close approximation is fine!
Now I have no reason to assume a close form even exists, and the expression looks quite ugly, so I would even go as far as to say that I highly doubt a closed form exists, but maybe someone can take a look and tell why it does/does not exist.
Thanks!
Edit: an other way to write the sequence is as:
$$a_n=(d-1)!\sum_{i=1}^{n}\binom{n-1-i+d}{n-2i}\binom{i+d-1}{i}$$
Maybe someone will recognize this or know how to deal with this.
Are you familiar with Gegenbauer polynomials ? Either way, we can use their definition
in rewriting the sum as $a_n=\pi\cdot\big(-{\bf i}\big)^n\cdot{\large\bf C}_n^{(d)}\bigg(\dfrac{\bf i}2\bigg)\cdot\displaystyle\lim_{\delta\to d}\dfrac{\csc\big(\delta\pi\big)}{\big(-\delta\big)!}$ , where the limit can
be evaluated using Euler's reflection formula for the $\Gamma$ function as $\dfrac{\Gamma\big(d\big)}\pi=\dfrac{\big(d-1\big)!}\pi$.