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
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Jun 23 2020
STATUS
approved