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

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.

Maple package

• EvenChange.txt, a Maple package to find generating functions, and explicit expressions as sums of quasi-polynomials, for answering the question in the title, as well as many other, more "serious" questions of interest that came up in Classical Invariant Theory.

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

• If you want to see generating functions, expressions as a sum of quasi-polynomials, the asympotitcs, and the value at n=googol (i.e. n=10¹⁰⁰) for the important sequence, number of ways of carrying altogether n coins in my two pockets (where each coin is either a penny, a nickel, a dime, or a quarter)

  the input file generates the output file.

• If you want to see generating functions, expressions as a sum of quasi-polynomials, the asympotitcs, and the value at n=googol (i.e. n=10¹⁰⁰) for the important sequence, number of ways of carrying altogether n coins in my two pockets (where each coin is either a penny, a nickel, a dime, a quarter, or a half-dollar)

  the input file generates the output file.

• If you want to see generating functions, expressions as a sum of quasi-polynomials, the asympotitcs, and the value at n=googol (i.e. n=10¹⁰⁰) for the important sequence, number of ways of carrying altogether n coins in my two pockets (where each coin is either a penny, a nickel, a dime, a quarter, a half-dollar, or a dollar coin)

  the input file generates the output file.

• If you want to see the generating functions ψ_(n)(t) for n from 2 to 18, (mentioned in Gert Almkvist's beautiful article "Invariants mostly old ones", that were computed for n up to n=4 by Faa de Bruno (in 1876) 10 and n=12 by James Joseph Sylvester and Fabian Franklin a few years later), and the value of the coefficient when n=10¹⁰⁰)

  the input file generates the output file.

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