A360708 Expansion of Sum_{k>=0} (x^2 / (1 - k*x))^k.

1, 0, 1, 1, 2, 5, 14, 42, 136, 479, 1825, 7433, 32053, 145608, 695081, 3479117, 18209842, 99373513, 563920590, 3320674902, 20255823092, 127799984935, 832807892861, 5597481205009, 38753768384761, 276057156622776, 2021100095469577, 15193591060371577

Offset: 0

Author

Seiichi Manyama, Feb 17 2023

Keywords

• nonn

Links

• Winston de Greef, Table of n, a(n) for n = 0..630

Crossrefs

Cf. A080108, A360709.

Cf. A000248, A360699.

Programs

Mathematica

Join[{1},Table[Sum[Binomial[n-k-1,k-1] * k^(n-2*k), {k,0,n/2}], {n,1,40}]] (* Vaclav Kotesovec, Feb 20 2023 *)

PARI

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (x^2/(1-k*x))^k))

PARI

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

Formula

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