Binomial transform
= Binomial transform
The binomial transform of a sequence $a_k$ is $b_n=\sum_{k=0}^n\binom nk a_k$. When $a_k$ counts objects on a specified set of $k$ positions, $b_n$ counts the choices obtained by selecting those positions from $n$ and filling them with such an object. Its inverse is $a_n=\sum_{k=0}^n(-1)^{n-k}\binom nk b_k$, by <binomial inversion>.