A242446 a(n) = Sum_{k=1..n} C(n,k) * k^(2*n).

1, 18, 924, 93320, 15609240, 3903974592, 1364509038592, 635177480713344, 379867490829555840, 283825251434680651520, 259092157573229145859584, 283735986144895532781391872, 367138254141051794797009309696, 554136240038549806366753446051840

Offset: 1

Author

Vaclav Kotesovec, May 14 2014

Keywords

• nonn
• easy

Comments

Generally, for p>=1, a(n) = Sum_{k=1..n} C(n,k) * k^(p*n) is asymptotic to sqrt(r/(p+r-p*r)) * r^(p*n) * n^(p*n) / (exp(p*n) * (1-r)^n), where r = p/(p+LambertW(p*exp(-p))).

Sum_{k=1..n} (-1)^(n-k) * C(n,k) * k^(p*n) = n! * stirling2(p*n,n).

Crossrefs

Cf. A007820, A072034, A242449, A256016.

Programs

Mathematica

Table[Sum[Binomial[n,k]*k^(2*n),{k,1,n}],{n,1,20}]

Formula

a(n) ~ sqrt(r/(2-r)) * r^(2*n) * n^(2*n) / (exp(2*n) * (1-r)^n), where r = 2/(2+LambertW(2*exp(-2))).