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A305922 Expansion of e.g.f. log(1 + 2*x/(exp(x) + 1)).

%I A305922 #7 Dec 04 2018 20:23:45
%S A305922 0,1,-2,5,-20,109,-738,5991,-56760,614601,-7486670,101330635,
%T A305922 -1508641140,24503026989,-431137315434,8169513007215,-165859346028656,
%U A305922 3591802533860497,-82644488286784326,2013441061219406739,-51777972823724776620,1401611202556240950645,-39838169568923591411810
%N A305922 Expansion of e.g.f. log(1 + 2*x/(exp(x) + 1)).
%C A305922 Logarithmic transform of A036968.
%H A305922 Alois P. Heinz, <a href="/A305922/b305922.txt">Table of n, a(n) for n = 0..430</a>
%H A305922 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A305922 Wikipedia, <a href="http://en.wikipedia.org/wiki/Genocchi_number">Genocchi number</a>
%H A305922 <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers</a>
%e A305922 E.g.f.: A(x) = x - 2*x^2/2! + 5*x^3/3! - 20*x^4/4! + 109*x^5/5! - 738*x^6/6! + ...
%p A305922 a:= proc(n) option remember; (t-> `if`(n=0, 0, t(n)-add(a(j)*j*
%p A305922       t(n-j)*binomial(n, j), j=1..n-1)/n))(i-> i*euler(i-1, 0))
%p A305922     end:
%p A305922 seq(a(n), n=0..25);  # _Alois P. Heinz_, Dec 04 2018
%t A305922 nmax = 22; CoefficientList[Series[Log[1 + 2 x/(Exp[x] + 1)], {x, 0, nmax}], x] Range[0, nmax]!
%t A305922 a[n_] := a[n] = n EulerE[n - 1, 0] - Sum[k Binomial[n, k] (n - k) EulerE[n - k - 1, 0] a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 0, 22}]
%t A305922 a[n_] := a[n] = 2 (1 - 2^n) BernoulliB[n] - Sum[k Binomial[n, k] 2 (1 - 2^(n - k)) BernoulliB[n - k] a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 0, 22}]
%Y A305922 Cf. A036968, A296837, A296838, A305711.
%K A305922 sign
%O A305922 0,3
%A A305922 _Ilya Gutkovskiy_, Jun 14 2018