<s>[INST] Do identities exist for the binomial series $\sum_{k=m+1}^{n+1} \binom{k}{m} \binom{n+1}{k-1} $? While examining the product of two upper triangular matrices, I've found that the $(m,n)$'th entry of the resulting matrix amounts to: $$\sum_{k=m+1}^{n+1} \binom{k}{m} \binom{n+1}{k-1} $$ when $n \geq m$ (all other entries are zero).
Although I have found some summations of products of binomial coefficients here, identities for the sum-product as described above have so far eluded me. Do you know whether identities for this series -- or perhaps even for generalizations of it -- exist?
 [/INST] $$\sum_{k}{{k\choose m} {n+1\choose k-1}}=\frac{1}{m}\sum_{k}k{k-1\choose m-1}{n+1\choose k-1} = \frac{n+1 \choose m-1}{m}\sum_{k}k{n-m+2 \choose k-m} = \frac{n+1 \choose m-1}{m}(\sum_{k}(k-m){n-m+2 \choose k-m}) + {n+1 \choose m-1}\sum_{k}{n-m+2 \choose k-m} = \frac{{n+1 \choose m-1}(n-m+2)}{m}(\sum_{k}{n-m+1 \choose k-m-1}) + {n+1 \choose m-1}2^{n-m+2} = \frac{{n+1 \choose m-1}2^{n-m+1}(n+m+2)}{m}$$
The identities I have used are ${n \choose k} = \frac{n}{k}{n-1 \choose k-1}, \sum_{k}{n \choose k} = 2^n, {a \choose b}{b \choose c} = {a \choose c}{a-c \choose b-c}$.
The summation in the question excludes $k=n+2,m$, so your answer is actually
$$\frac{{n+1 \choose m-1}2^{n-m+1}(n+m+2)}{m}-{n+2\choose m}-{n+1\choose m-1}   $$
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