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A357240 Expansion of e.g.f. 2 * (exp(x) - 1) / (exp(exp(x) - 1) + 1).

%I A357240 #16 Jun 23 2023 18:22:34
%S A357240 0,1,0,-2,-5,-4,32,225,794,190,-22291,-200298,-920244,924223,65848880,
%T A357240 716920754,3831260555,-13147083976,-575844827780,-7162425813919,
%U A357240 -40755845041730,320194436283162,11810647258173653,161108090793013130,896865861205240824,-14305712791762925929,-487306962045115504436
%N A357240 Expansion of e.g.f. 2 * (exp(x) - 1) / (exp(exp(x) - 1) + 1).
%C A357240 Stirling transform of the Genocchi numbers (of first kind, A036968).
%H A357240 Alois P. Heinz, <a href="/A357240/b357240.txt">Table of n, a(n) for n = 0..494</a>
%F A357240 a(n) = 2 * Sum_{k=0..n} Stirling2(n,k) * (1 - 2^k) * Bernoulli(k).
%F A357240 a(n) ~ Pi^(3/2) * 2^(n + 7/2) * n^(n + 1/2) * (cos(n*arctan(2*arctan(Pi)/log(1 + Pi^2))) * (Pi*log(1 + Pi^2) + 2*arctan(Pi)) + (log(1 + Pi^2) - 2*Pi*arctan(Pi)) * sin(n*arctan(2*arctan(Pi)/log(1 + Pi^2)))) / ((1 + Pi^2) * exp(n) * (4*arctan(Pi)^2 + log(1 + Pi^2)^2)^(n/2 + 1)). - _Vaclav Kotesovec_, Oct 04 2022
%p A357240 b:= proc(n, m) option remember; `if`(n=0, `if`(m=0, 0,
%p A357240       m*euler(m-1, 0)), m*b(n-1, m)+b(n-1, m+1))
%p A357240     end:
%p A357240 a:= n-> b(n, 0):
%p A357240 seq(a(n), n=0..26);  # _Alois P. Heinz_, Jun 23 2023
%t A357240 nmax = 26; CoefficientList[Series[2 (Exp[x] - 1)/(Exp[Exp[x] - 1] + 1), {x, 0, nmax}], x] Range[0, nmax]!
%t A357240 Table[2 Sum[StirlingS2[n, k] (1 - 2^k) BernoulliB[k], {k, 0, n}], {n, 0, 26}]
%o A357240 (PARI) a(n) = 2*sum(k=0, n, stirling(n, k, 2)*(1-2^k)*bernfrac(k)); \\ _Michel Marcus_, Sep 20 2022
%Y A357240 Cf. A001469, A003149, A036968, A059371.
%K A357240 sign
%O A357240 0,4
%A A357240 _Ilya Gutkovskiy_, Sep 19 2022