Questions tagged [combinatorial-identities]

I stumbled upon a curious identity that seems to hold for all integers $n\ge3$: $$\sum_{i=0}^{\lfloor n/3\rfloor}\prod_{j=1}^i{-{n-3j\choose3}\over{n\choose3}-{n-3j\choose3}} ={n\over3}\sum_{i=0}^{\...

How this can be proved? $$ E = {}_2F_1(-\frac{1}{2}, \frac{1}{3}, \frac{4}{3},-8) = \frac{6}{5} - \frac{\chi}{2} $$ where $$ \chi = \frac{6\sqrt{\pi}}{5}\frac{\Gamma(\frac{1}{3})}{\Gamma(-\frac{1}{6})}...

In this interview, Ira Gessel mentions the following results: Result 1: Let $B_n$ denote the $n^{\text{th}}$ Bernoulli number. Define the series $$B(x) = \sum_{n=2}^{\infty} \frac{B_nx^{n-1}}{n(n-1)}.$...

I came across the following identity in my research: $$ \sum_{m=0}^s \frac{(-1)^m (a+2m)}{m!(s-m)! (a+m)_{s+1}}=\delta_{s,0} $$ where $(a)_n= a(a+1)\cdots (a+n-1)$ is the Pochhammer symbol. One can ...

Consider the well-known sequence $C_k=\frac1{k+1}\binom{2k}k$ of Catalan numbers. I came across the below identity while working with certain generating functions. I thought it might be of interest to ...