Mersenne Binomials and the Coefficients of the non-associative Exponential

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dc.creatorGerritzen, Lothar
dc.date2005-02-17
dc.date.accessioned2026-07-07T05:17:07Z
dc.date.available2026-07-07T05:17:07Z
dc.descriptionThe 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 $$\sum \hat{a}(t) = ω(n)$$ where the sum is extended over all trees $t$ in $M$ of degree $n$ and $$ω(n) = \frac{2^{n-1}}{n!} \prod^{n-1}_{i=1} (2^i - 1).$$ The prime factorization of $ω(n)$ is described. The sequence $(ω(n))_{n \ge 1}$ seems to be of interest.
dc.description12 pages
dc.identifierhttps://arxiv.org/abs/math/0502379
dc.identifierhttp://arxiv.org/abs/math/0502379
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/74233
dc.subjectRings and Algebras
dc.subject17A50
dc.titleMersenne Binomials and the Coefficients of the non-associative Exponential
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