# Generating Function for Natural Numbers/Corollary

## Theorem

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

$\forall n \in \N_{> 0}: a_n = n - 1$

That is:

$\sequence {a_n} = 1, 2, 3, 4, \ldots$

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

$H \paren z = \dfrac z {\paren {1 - z}^2}$

## Proof

$\sequence {a_n} = 0, 1, 2, 3, 4, \ldots$

has the generating function:

$G \paren z = \dfrac 1 {\paren {1 - z}^2}$
$z G \paren z = \dfrac z {\paren {1 - z}^2}$

is the generating function for the sequence defined as:

$\forall n \in \N_{> 0}: a_{n - 1} = n - 1$

that is:

$0, a_1, a_2, \ldots$

where:

$a_1 = 1, a_2 = 2, \ldots$

That is:

$\sequence {a_n} = 1, 2, 3, 4, \ldots$

$\blacksquare$