A384028 a(n) = Sum_{k=0..2*n} Stirling1(2*n+1, 2*n+1-k) * Stirling1(2*n+1, k+1).

1, 13, 2273, 1184153, 1251320145, 2232012515445, 6032418472347265, 23007314730623658225, 117745011140615270168865, 778780810721500176081199325, 6466413475830749109197652489569, 65861328745485785925705177696147337, 807448787241269228642562251336079833585

Offset: 0

Author

Vaclav Kotesovec, May 17 2025

Keywords

• nonn

Crossrefs

Cf. A129256, A234324.

Programs

Mathematica

Table[Sum[StirlingS1[2*n+1, 2*n+1-j]*StirlingS1[2*n+1, j+1], {j, 0, 2*n}], {n, 0, 15}]

Formula

a(n) ~ 2^(6*n) * w^(4*n + 3/2) * n^(2*n - 1/2) / (sqrt(Pi*(w-1)) * exp(2*n) * (2*w-1)^(2*n)), where w = -LambertW(-1, -exp(-1/2)/2) = 1.756431208626169676982737616...

a(n) = A129256(2*n) = [x^(2*n)] Product_{k=0..2*n} (1 + k*x)^2. - Seiichi Manyama, May 17 2025