# Doubly Sequenced Generating Function for Binomial Coefficients

## Theorem

Let $\left \langle {a_{m n} }\right \rangle$ be the doubly subscripted sequence defined as:

$\forall m, n \in \N_{\ge 0}: a_{m n} = \dbinom n m$

where $\dbinom n m$ denotes a binomial coefficient.

Then the generating function for $\left \langle {a_{m n} }\right \rangle$ is given as:

$G \left({w, z}\right) = \dfrac 1 {1 - z - w z}$

## Proof

 $\displaystyle G \left({w, z}\right)$ $=$ $\displaystyle \sum_{m, \, n \mathop \ge 0} a_{m n} w^m z^n$ $\quad$ Definition of Generating Function for Doubly Subscripted Sequence $\quad$ $\displaystyle$ $=$ $\displaystyle \sum_{m, \, n \mathop \ge 0} \dbinom n m w^m z^n$ $\quad$ $\quad$ $\displaystyle$ $=$ $\displaystyle \sum_{n \mathop \ge 0} \left({1 + w}\right)^n z^n$ $\quad$ Binomial Theorem $\quad$ $\displaystyle$ $=$ $\displaystyle \sum_{n \mathop \ge 0} \left({\left({1 + w}\right) z}\right)^n$ $\quad$ $\quad$ $\displaystyle$ $=$ $\displaystyle \dfrac 1 {1 - \left({1 + w}\right) z}$ $\quad$ Sum of Infinite Geometric Progression $\quad$ $\displaystyle$ $=$ $\displaystyle \dfrac 1 {1 - z - wz}$ $\quad$ $\quad$

$\blacksquare$