## Interesting Combinatorial Sum

**Point of post:** This is just an interesting combinatorial sum I solved on AOPS a few minutes ago. I mention it because, while sometimes less elegant, one must remember to never disregard “continuous” math even when one is working in “discrete” areas.

**Problem:** For such that prove that

**Proof: **The first thing one might notice is that

so that

But, rearranging this we arrive at

which equals

but appealing to the Binomial Theorem we see that the above is equal to

where is the Beta Function. But, appealing to an alternative form of the Beta function we see that the sum is equal to

where is the Gamma Function.

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