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 $$\hat{a}(t) := \frac{a(t)}{2^{n-1}\cdot \prod^{n-1}_{i=1}(2^i - 1)}$$ 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 $$\omega(n) = \frac{2^{n-1}}{n!} \prod^{n-1}_{i=1} (2^i - 1).$$ The prime factorization of $\omega(n)$ is described. The sequence $(\omega(n))_{n \ge 1}$ seems to be of interest.