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7.4:  Generating Functions

$g(x) = h_0 + h_1x +  h_2x^2 + ....$ is the {\it generating function} for the sequence $h_0, h_1, h_2, ....$.


Ex:  The generating fn for the sequence 2, 3, 4, 0, 0, 0, ... is

$g(x) = 2 + 3x + 4x^2$


Ex:  The generating function for the sequence 1, 1, 1, ... is

$g(x) = 1 + x + x^2 + ... = {1 \over 1 - x}$


Ex:  The generating function for the sequence 

$\left(\matrix{m \cr 0 }\right)$,
$\left(\matrix{m \cr 1 }\right)$,
$\left(\matrix{m \cr 2 }\right)$,
...,
$\left(\matrix{m \cr m }\right)$ is

$g(x) = \left(\matrix{m \cr 0 }\right) + \left(\matrix{m \cr 1 }\right)x + \left(\matrix{m \cr 2 }\right)x^2 + ... \left(\matrix{m \cr m }\right)x^m= (1 + x)^m$



Ex:  Suppose $\alpha \in {\cal R}$.  The generating function for the sequence 

$\left(\matrix{\alpha \cr 0 }\right)$,
$\left(\matrix{\alpha \cr 1 }\right)$,
$\left(\matrix{\alpha \cr 2 }\right)$,
...,
 is

$g(x) = \left(\matrix{\alpha \cr 0 }\right) + \left(\matrix{\alpha \cr 1 }\right)x + \left(\matrix{\alpha \cr 2 }\right)x^2 + ... = (1 + x)^\alpha$

Ex:  Let $h_n$ = number of nonnegative solutions to \hfil \break
 $e_1 + e_2 + ... + e_k = n$

Thus $h_n = $

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Thus $g(x) = $











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