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
From Generating Function for Natural Numbers:


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

has the generating function:
 * $G \paren z = \dfrac 1 {\paren {1 - z}^2}$

Then by Generating Function by Power of Parameter:
 * $z G \paren z = \dfrac z {\paren {1 - z}^2}$

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