In How many ways can I carry a total of n coins in my two pockets, and have the same amount in both pockets?

By Shalosh B. Ekhad and Doron Zeilberger

.pdf   .ps   .tex  

(Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger and )

In fond memory of Gert Almkvist (April 17, 1934 - Nov. 24, 2018)

Posted: Jan. 23, 2019

In Gert Almkvist's beautiful article, entitled "Invariants, mostly old ones", (that appeared in the Pacific Journal of Mathematics, vol. 86 (1980), pp. 1-13) he talked about a sequence of generating functions that came up in his work (some of it in collaboration with Robert Fossum), that turned out to be the same as generating functions for the number of covariants of binary quadratic forms studied by Faa de Bruno, Cayley, Sylvester, and other 19th century savants. Using a very simple-minded Maple program (that uses the partial-fraction decomposition of a rational function), we recompute them, and go all the way to degree 18.

It turns out that the same method can be used to answer lots of other combinatorial questions, including the one in the title.

Added Feb. 1, 2019: The sequence of the title is now OEIS sequence A323825

Maple package

Sample Input and Output Files for the Maple package EvenChange.txt,

Personal Journal of Shalosh B. Ekhad and Doron Zeilberger

Doron Zeilberger's Home Page