Mersenne Binomials and the Coefficients of the non-associative Exponential

Abstract

The non-associative exponential series $exp(x)$ is a power series with monomials from the magma $M$ of finite, planar rooted trees. The coefficient $a(t)$ of $exp(x)$ relative to a tree $t$ of degree $n$ is a rational number and it is shown that $$\hat{a}(t):=\frac{a(t)}{2^{n-1}\cdot {\prod }_{i=1}^{n-1}(2^i-1)}$$ is an integer which is a product of Mersenne binomials. One obtains summation formulas $$\sum \hat{a}(t)=\omega (n)$$ where the sum is extended over all trees $t$ in $M$ of degree $n$ and $$\omega (n)=\frac{2^{n-1}}{n!}\prod _{i=1}^{n-1}(2^i-1).$$ The prime factorization of $\omega (n)$ is described. The sequence $(\omega (n){)}_{n\ge 1}$ seems to be of interest.

12 pages