On a Luschny question

Abstract

Let $E_n(x)$ be Euler polynomial, ${\nu }_2(n)$ be $2-$adic order of $n,$ $\{g(n)\}$ be the characteristic sequence for $\{2^n-1{\}}_{n\ge 1}.$ Recently Peter Luschny asked (cf. \cite{5}, sequence A290646): is for $n\ge 1$ A290646=A135517? According to A135517, A091090 and a formula there, this question is equivalent to the following one: is, for $n\ge 1,$ the denominator of $E_n(x)-E_n(1)$ equal to $2^{{\nu }_2(n+1)-g(n)}$? In this note we answer in the affirmative on this question.

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