6.3b Hockey-Stick Identity

Hockey-Stick Identity

The Hockey-Stick Identity is a property of the binomial coefficient, stating that the sum of entries along a diagonal in Pascal’s triangle equals the entry directly below and one place to the right of the last term in the sum.

Formally, for integers $n\ge r\ge 0$, the identity is expressed as
$\left(\genfrac{}{}{0ex}{}{n}{k+1}\right)={\sum }_{i=1}^{n-k}\left(\genfrac{}{}{0ex}{}{n-i}{k}\right).$

Proof