I recently came across a complicated sum while working on my homework. Usually I have no issues evaluating sums, but this one has stumped me. WolframAlpha managed to find a closed form solution, but I can't seem to work out how I should go about deriving it. Here's the sum and closed form from WolframAlpha:
I have tried writing out the sum term by term to look for patterns. I noticed the factorials in the denominator seem to pair up sometimes, but I haven't been able to leverage that to any use. The main way I know to handle factorials in sums is to find a power series representation that matches the sum, but this sum doesn't seem to match anything I can find.
I appreciate any hints or solutions. Ideally, a solution without any prior knowledge of the result.