Web Reference: Our goal now is to gather some tools to build the generating function of a particular given sequence. Let's see what the generating functions are for some very simple sequences. The simplest of all: 1, 1, 1, 1, 1, …. What does the generating series look like? It is simply \ (1 + x + x^2 + x^3 + x^4 + \cdots\text {.}\) Generating Functions - Introduction Example Given a sequence {an: n ≥ 0} of real numbers, the “function” = σ ∞ is called the generating =0 function of the sequence. This function may or may not have meaning in the sense of an infinite series like you studied in calculus classes. In this section we give formulas for generating functions enumerating the sequence {fan + b} given an ordinary generating function F(z), where a ≥ 2, 0 ≤ b < a, and a and b are integers (see the main article on transformations).
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