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

## TL;DR

This paper explores a combinatorial problem of counting ways to split n coins equally into two pockets, using generating functions and computational methods to extend previous mathematical results.

## Contribution

It introduces a simple computational approach with Maple to enumerate solutions to the coin-splitting problem and related combinatorial questions, extending known results to degree 18.

## Key findings

- Generated new sequences for the coin-splitting problem
- Demonstrated the effectiveness of partial-fraction decomposition in enumeration
- Extended previous mathematical results to higher degrees

## Abstract

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, 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 savantes. 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 many other enumeration questions, including the one in the title.