A068942 a(n) = Bo(n^2), n=0,1..., where Bo(n) are the ordered Bell numbers, A000670.

1, 1, 75, 7087261, 5315654681981355, 106697365438475775825583498141, 144199280951655469628360978109406917583513090155, 27656793065414932606012896651489726461435178241015434306518713649426461

Offset: 0

Author

Karol A. Penson, Mar 09 2002

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..20

Crossrefs

Cf. A000670, A068939, A249938, A249941.

Programs

Mathematica

a[n_] := PolyLog[-n^2, 1/2]/2; a[0] = 1; Table[a[n], {n, 0, 7}] (* Jean-François Alcover, Mar 30 2016 *)
Table[Sum[k!*StirlingS2[n^2, k], {k, 0, n^2}], {n, 0, 10}] (* Vaclav Kotesovec, Jun 08 2021 *)

PARI

a(n) = sum(k=0, n^2, k!*stirling(n^2, k, 2)); \\ Seiichi Manyama, Jan 17 2022

Formula

a(n) = Sum_{k>=1} (k^(n^2))/2^(k+1); this is the analog of the Dobinski formula.

Integral representation as n-th moment of a positive function on a positive half-axis, in Maple notation: a(n)=int(x^n*(sum(exp(-ln(x)^2/(4*ln(k))) / (2^k*sqrt(ln(k))), k=2..infinity)/(4*sqrt(Pi)*x)+Dirac(x-1)/4), x=0..infinity).

a(n) ~ (n^2)! / (2 * log(2)^(n^2 + 1)). - Vaclav Kotesovec, Jun 08 2021