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
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(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. 113) 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 simpleminded Maple program (that uses the partialfraction 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

EvenChange.txt,
a Maple package to find generating functions, and explicit expressions as sums of quasipolynomials, 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 quasipolynomials, the asympotitcs, and
the value at n=googol (i.e. n=10^{100}) 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 quasipolynomials, the asympotitcs, and
the value at n=googol (i.e. n=10^{100}) 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 halfdollar)
the input file generates the
output file.

If you want to see generating functions, expressions as a sum of quasipolynomials, the asympotitcs, and
the value at n=googol (i.e. n=10^{100}) 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 halfdollar, 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=10^{100})
the input file generates the
output file.
Personal Journal of Shalosh B. Ekhad and Doron Zeilberger
Doron Zeilberger's Home Page