Journal of Integer Sequences, Vol. 21 (2018), Article 18.3.3

A Combinatorial Proof for the Generating Function of Powers of a Second-Order Recurrence Sequence

Yifan Zhang and George Grossman
Department of Mathematics
Central Michigan University
Mount Pleasant, MI 48858


In this paper, we derive a formula for the generating function of powers of a second-order linear recurrence sequence, with initial conditions 0 and 1. As an example, we find the generating function of the powers of the nonnegative integers. We also find new formulas for computing Eulerian polynomials.

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(Concerned with sequences A000032 A000045 A000129 A000290 A000578 A000583 A000584 A001014 A001015 A001016 A001017 A001045 A001477 A001582 A007598 A008292 A008454 A030186 A056570 A056571 A056572 A056573 A056574 A056585 A056586 A056587 A079291 A110272 A139818.)

Received January 17 2017; revised versions received February 3 2017; November 1 2017; January 21 2018. Published in Journal of Integer Sequences, March 9 2018.

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