A192564 a(n) = Sum_{k=0..n} abs(Stirling1(n,k))*Stirling2(n,k)*(k!)^2.

1, 1, 5, 74, 2186, 106524, 7703896, 773034912, 102673179360, 17429291711280, 3680338415133024, 945958227345434016, 290761516548473591232, 105309706114422166775040, 44384982810939832477305600, 21536846291826596564956445184

Offset: 0

Author

Emanuele Munarini, Jul 04 2011

Keywords

• nonn

Crossrefs

Cf. A047793, A048144, A382792.

Programs

Mathematica

Table[Sum[Abs[StirlingS1[n,k]]StirlingS2[n,k]k!^2,{k,0,n}],{n,0,100}]
nmax = 20; Table[SeriesCoefficient[1/(1 + (E^x - 1)*Log[1 - y]), {x, 0, n}, {y, 0, n}], {n, 0, nmax}] * Range[0, nmax]!^2 (* Vaclav Kotesovec, Apr 08 2025 *)

Maxima

makelist(sum(abs(stirling1(n,k))*stirling2(n,k)*k!^2,k,0,n),n,0,24);

Formula

a(n) ~ c * LambertW(-1, -r*exp(-r))^n * n!^2 / (sqrt(n) * LambertW(-exp(-1/r)/r)^n), where r = 0.673313285145753168... is the root of the equation (1 + 1/(r*LambertW(-exp(-1/r)/r))) * (r + LambertW(-1, -r*exp(-r))) = 1 and c = 0.27034346270211507329954765593360596752557904498770241464597402478625037569... - Vaclav Kotesovec, Jul 05 2021

a(n) = (n!)^2 * [(x*y)^n] 1 / (1 + (exp(x) - 1) * log(1 - y)). - Ilya Gutkovskiy, Apr 06 2025