How to Count by Robert A. Beeler

Author:Robert A. Beeler
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Since λ i is a root of multiplicity m i for , it follows that:

Suppose that R(x) is the generating function associated with the sequence R n . In other words,

We now manipulate the series to take advantage of the recurrence.

Define . Further, for we have that . Thus,

where and are polynomials of degree at most k − 1. We now re-index the summations using the change of variables . This gives us

Since the variable used to index the summation is irrelevant, we have that

The characteristic polynomial factors as

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