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

0,3

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

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

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 *)

Crossrefs

Cf. A001250, A317022.

Sequence in context: A356291 A001339 A307030 * A012316 A261600 A331617

Adjacent sequences: A335785 A335786 A335787 * A335789 A335790 A335791

Keyword

nonn

Author

Geoffrey Critzer, Jun 23 2020

Status

approved