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A305711 Expansion of e.g.f. exp(2*x/(exp(x) + 1)).

%I A305711 #7 Mar 27 2019 03:53:12
%S A305711 1,1,0,-2,-1,11,13,-111,-220,1756,5051,-39775,-153191,1215345,5952668,
%T A305711 -48020714,-288569149,2377190003,17069110381,-143857868895,
%U A305711 -1209439895944,10435153277620,101078662547567,-892827447251575,-9834570608359487,88900938146195601,1101567283699652888
%N A305711 Expansion of e.g.f. exp(2*x/(exp(x) + 1)).
%H A305711 Wikipedia, <a href="http://en.wikipedia.org/wiki/Genocchi_number">Genocchi number</a>
%H A305711 <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers</a>
%e A305711 exp(2*x/(exp(x) + 1)) = 1 + x - 2*x^3/3! - x^4/4! + 11*x^5/5! + 13*x^6/6! - 111*x^7/7! - 220*x^8/8! + ...
%p A305711 a:=series(exp(2*x/(exp(x)+1)),x=0,27): seq(n!*coeff(a,x,n),n=0..26); # _Paolo P. Lava_, Mar 26 2019
%t A305711 nmax = 26; CoefficientList[Series[Exp[2 x/(Exp[x] + 1)], {x, 0, nmax}], x] Range[0, nmax]!
%t A305711 a[n_] := a[n] = Sum[k EulerE[k - 1, 0] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 26}]
%t A305711 a[n_] := a[n] = Sum[2 (1 - 2^k) BernoulliB[k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 26}]
%Y A305711 Cf. A036968, A296835, A296836.
%K A305711 sign
%O A305711 0,4
%A A305711 _Ilya Gutkovskiy_, Jun 08 2018