A355471 Expansion of Sum_{k>=0} (x/(1 - k^2 * x))^k.

1, 1, 2, 10, 77, 808, 11257, 196072, 4136897, 103755904, 3034193921, 101901347944, 3885951145969, 166605168800704, 7961498177012993, 420976047757358776, 24475992585921169553, 1556007778666449968128, 107625967130820901112833

Offset: 0

Author

Seiichi Manyama, Jul 03 2022

Keywords

• nonn

Crossrefs

Cf. A080108, A135746, A234568, A355463, A355472.

Programs

Mathematica

Flatten[{1, Table[Sum[Binomial[n-1,k-1] * k^(2*(n-k)), {k,1,n}], {n,1,20}]}] (* Vaclav Kotesovec, Feb 16 2023 *)

PARI

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (x/(1-k^2*x))^k))

PARI

a(n) = if(n==0, 1, sum(k=1, n, k^(2*(n-k))*binomial(n-1, k-1)));

Formula

a(n) = Sum_{k=1..n} k^(2*(n-k)) * binomial(n-1,k-1) for n > 0.