# Generating Function for Triangular Numbers

## Theorem

Let $T_n$ denote the $n$th triangular number.

Then the generating function for $\sequence {T_n}$ is given as:

$\ds \map G z = \frac z {\paren {1 - z}^3}$

### Corollary

Let $\sequence {b_n}$ be the sequence defined as:

$\forall n \in \N_{> 0}: b_n = \dfrac {\paren {n + 1} \paren {n + 2} } 2$

That is:

$\sequence {b_n}_{n \mathop \ge 0} = 1, 3, 6, 10, \ldots, \dbinom {n + 2} 2, \ldots$

Then the generating function for $\sequence {b_n}$ is given as:

$H \paren z = \dfrac 1 {\paren {1 - z}^3}$

## Proof

 $\ds \frac z {\paren {1 - z}^3}$ $=$ $\ds z \paren {1 - z}^{-3}$ Exponent Combination Laws for Negative Power $\ds$ $=$ $\ds z \sum_{n \mathop = 0}^\infty \dbinom {-3} n \paren {-z}^n$ General Binomial Theorem $\ds$ $=$ $\ds z \sum_{n \mathop = 0}^\infty \paren {-1}^n \dbinom {n + 2} n \paren {-z}^n$ Negated Upper Index of Binomial Coefficient $\ds$ $=$ $\ds z \sum_{n \mathop = 0}^\infty \dbinom {n + 2} n z^n$ $\ds$ $=$ $\ds z \sum_{n \mathop = 0}^\infty \dbinom {n + 2} 2 z^n$ Symmetry Rule for Binomial Coefficients $\ds$ $=$ $\ds \sum_{n \mathop = 0}^\infty \dbinom {n + 2} 2 z^{n + 1}$ $\ds$ $=$ $\ds \sum_{n \mathop = 1}^\infty \dbinom {n + 1} 2 z^n$ $\ds$ $=$ $\ds \sum_{n \mathop = 1}^\infty T_n z^n$ Corollary to Binomial Coefficient with Two $\ds$ $=$ $\ds 0 z^0 + \sum_{n \mathop = 1}^\infty T_n z^n$ $\ds$ $=$ $\ds \sum_{n \mathop = 0}^\infty T_n z^n$ $T_0 = 0$ by Definition 1 of Triangular Number $\ds$ $=$ $\ds \map G z$ Definition of Generating Function

$\blacksquare$