Evaluate the sum $\sum_{i=0}^n \binom{n}{i}^2 i^k$

Asked Mar 03 '16 at 17:35

Active Mar 03 '16 at 17:35

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Given a positive integer $k$, is it possible to evaluate the following sum? $$ \sum_{i=0}^n \binom{n}{i}^2 i^k\,\,\,? $$

[I know just for $k=0$ the sum is $\binom{2n}{n} \approx 4^n/\sqrt{n}$..]

asked Mar 03 '16 at 17:35

$f(n,m)=\sum_{i=0}^n \binom ni^2 i^{m-2}i^2=n^2\sum_{i=0}^{n-1}\binom{n-1}{i-1}^2i^{m-2}$
$\implies f(n,m)=n^2f(n-1,m-2)$
– lab bhattacharjee Mar 03 '16 at 17:49
why the second sum goes up to n-1? – Mar 03 '16 at 17:53
$$f(n,m)=\sum_{i=0}^n \binom ni^2 i^{m-2}i^2=n^2\sum_{i-1=0}^{n-1}\binom{n-1}{i-1}^2i^{m-2}$$
$$\implies f(n,m)=n^2f(n-1,m-2)$$
– lab bhattacharjee Mar 03 '16 at 17:58
Oh, ok.. so, we miss only the case $k=1$, right? – Mar 03 '16 at 18:04

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