Generating Function for Triangular Numbers
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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 |
$\blacksquare$