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

1, 1, 3, 11, 49, 263, 1675, 12417, 105183, 1002475, 10616589, 123679907, 1571831251, 21640964933, 320872742611, 5097445680435, 86377624918593, 1555173730665199, 29646960589439139, 596571563234557361, 12636340495630310359

Offset: 0

Author

Geoffrey Critzer, Jun 23 2020

Keywords

• nonn

Comments

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.

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..199

Crossrefs

Cf. A001250, A317022.

Programs

Mathematica

nn = 20; a[x_] := Tan[x] + Sec[x]; b[x_] := 2 a[x] - 1 - x;
Range[0, nn]! CoefficientList[Series[b[Exp[x] - 1], {x, 0, nn}], x]
(* Second program: *)
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 *)

Formula

a(n) = Sum_{k=1..n} Stirling2(n,k)*A001250(k).

E.g.f.: B(exp(x)-1) where B(x) = 2(tan(x) + sec(x))-1-x.

a(n) ~ 8 * n! / ((Pi+2) * log(1 + Pi/2)^(n+1)). - Vaclav Kotesovec, Jun 24 2020