1
vote
0answers
134 views

n-th self discrete convolution

Lets define discrete $ f_N(i) = 1,\space i = 1...N $ I need to find $ G_N^m = \underbrace {f_N * f_N * ... * f_N}_{m} $ For example $G_6^3$ have value (1,3,6,10,15,21,25,27,27,25,21,15,10,6,3,1) , ...
4
votes
3answers
348 views

Alternating sign Vandermonde convolution

The well-known Vandermonde convolution gives us the closed form $\sum_{k=0}^n {r\choose k}{s\choose n-k} = {r+s \choose n}$. For the case $r=s$, it is also known that $\sum_{k=0}^n (-1)^k {r \choose ...
13
votes
4answers
673 views

How this operation is called?

This operation is similar to discrete convolution and cross-correlation, but has binomial coefficients: $$f(n)\star g(n)=\sum_{k=0}^n \binom{n}{k}f(n-k)g(k) $$ Particularly, $$a^n\star ...