Numbers in the base $e^\pi$

Abstract

A large-scale experiment was conducted to find formulas relating to the base $e^\pi$. The numbers in this base are $$x = \sum_{n=0}^\infty {a(n)\over e^{\pi n}}$$ where $a(n)$ is taken from the OEIS catalog. These experiments were inspired by several facts. Indeed, it is known that the formula generating the partitions of integers is generated by an infinite product $$\prod_{k\ge1}^\infty {1\over 1-x^k} = \sum_{n=0}^\infty p(n)x^n$$ that when evaluated at $x=e^{-\pi}$ is equal to $${2^{3/8} \Gamma(3/4) \over \pi^{1/4} e^{\pi/24}}\ .\qquad\qquad (1)$$ By analyzing the 387500 sequences of the OEIS catalog, the model that was used is based on the fact that the infinite sum evaluated at $e^\pi$, is an expression that can be detected using a program like lindep from Pari-Gp. The process made it possible to find 793 expresssions similar to (1).