Journal of Integer Sequences, Vol. 22 (2019), Article 19.3.2

Polynomial Analogues of Restricted b-ary Partition Functions

Karl Dilcher
Department of Mathematics and Statistics
Dalhousie University
Halifax, NS B3H 4R2

Larry Ericksen
P. O. Box 172
Millville, NJ 08332-0172


Given an integer b ≥ 2, a well studied concept of a b-ary partition function of a positive integer n counts the number of representations of n as sums of powers of b, with each power occurring up to λ times, for a fixed λ ≥ 1. In this paper we introduce and study a multivariable polynomial sequence that reduces to the restricted b-ary partition function when all variables are taken to be 1. In particular, we show that this polynomial sequence characterizes all restricted b-ary partitions for each n, generalizing previous results on hyperbinary and hyper b-ary representations. All this will follow from considering more general concepts of restricted b-ary partition functions.

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(Concerned with sequences A002487 A018819.)

Received December 2 2018; revised version received May 8 2019. Published in Journal of Integer Sequences, May 17 2019.

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