A384496 a(n) = Sum_{k=0..n} binomial(n,k)^3 * abs(Stirling1(2*k,k)) * abs(Stirling1(2*n-2*k,n-k)).

1, 2, 30, 1044, 68474, 7180900, 1050625720, 196205015216, 44361477901818, 11751610490415828, 3567182462164189140, 1220655384720089761080, 464932034143270233958352, 195108754505934104188716064, 89452431045403310104416682304, 44489455448017524780072427344000

Offset: 0

Author

Vaclav Kotesovec, May 31 2025

Keywords

• nonn

Comments

In general, for m > 1, Sum_{k=0..n} binomial(n,k)^m * abs(Stirling1(2*k,k)) * abs(Stirling1(2*n-2*k,n-k)) ~ 2^((m+2)*n + (m-3)/2) * n^(n - (m+1)/2) * w^(2*n) / (sqrt(m-1) * (w-1) * Pi^((m+1)/2) * exp(n) * (2*w-1)^n), where w = -LambertW(-1, -exp(-1/2)/2).

Crossrefs

Cf. A187656 (m=0), A187658 (m=1), A384495 (m=2), A384472.

Programs

Mathematica

Table[Sum[Binomial[n,k]^3 * Abs[StirlingS1[2*k,k]] * Abs[StirlingS1[2*n-2*k,n-k]], {k, 0, n}], {n, 0, 20}]

Formula

a(n) ~ 2^(5*n - 1/2) * n^(n-2) * w^(2*n) / ((w-1) * Pi^2 * exp(n) * (2*w-1)^n), where w = -LambertW(-1, -exp(-1/2)/2) = 1.7564312086261696769827376166...