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A335788 Expansion of e.g.f. 2*sec(exp(x)-1) - 2*tan(exp(x)-1) - exp(x).

%I A335788 #18 Aug 02 2021 17:44:10
%S A335788 1,1,3,11,49,263,1675,12417,105183,1002475,10616589,123679907,
%T A335788 1571831251,21640964933,320872742611,5097445680435,86377624918593,
%U A335788 1555173730665199,29646960589439139,596571563234557361,12636340495630310359
%N A335788 Expansion of e.g.f. 2*sec(exp(x)-1) - 2*tan(exp(x)-1) - exp(x).
%C A335788 a(n) is the number of ways to partition {1,2,...,n} into any number of blocks, then order the blocks so that the set of least elements of the blocks is an alternating permutation.
%H A335788 Michael De Vlieger, <a href="/A335788/b335788.txt">Table of n, a(n) for n = 0..199</a>
%F A335788 a(n) = Sum_{k=1..n} Stirling2(n,k)*A001250(k).
%F A335788 E.g.f.: B(exp(x)-1) where B(x) = 2(tan(x) + sec(x))-1-x.
%F A335788 a(n) ~ 8 * n! / ((Pi+2) * log(1 + Pi/2)^(n+1)). - _Vaclav Kotesovec_, Jun 24 2020
%t A335788 nn = 20; a[x_] := Tan[x] + Sec[x]; b[x_] := 2 a[x] - 1 - x;
%t A335788 Range[0, nn]! CoefficientList[Series[b[Exp[x] - 1], {x, 0, nn}], x]
%t A335788 (* Second program: *)
%t A335788 Array[Abs[-1 + Sum[4 StirlingS2[#, k] Abs[PolyLog[-k, I]], {k, #}]] &, 21, 0] (* _Michael De Vlieger_, Aug 02 2021, after _Jean-François Alcover_ at A001250 *)
%Y A335788 Cf. A001250, A317022.
%K A335788 nonn
%O A335788 0,3
%A A335788 _Geoffrey Critzer_, Jun 23 2020