 Binomial transform

The binomial transform of a sequence $a_k$ is $b_n={\sum }_{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)a_k$. 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 $a_n={\sum }_{k=0}^n(-1{)}^{n-k}(\genfrac{}{}{0ex}{}{n}{k})b_k$, by binomial inversion.

The Binomial transform is a mathematical operation that transforms a sequence of numbers into another sequence through a series of binomial coefficients. It is particularly useful in combinatorics and has applications in various areas of mathematics, including generating functions and number theory.