A360479 Expansion of Sum_{k>=0} (x * (1 + (k * x)^2))^k.

1, 1, 1, 2, 9, 28, 81, 369, 1753, 7323, 36337, 207401, 1114345, 6308368, 40326033, 256982157, 1658573497, 11650405774, 83966740913, 608348063576, 4659734909385, 36973835868521, 295709600709585, 2454457098977559, 21106884235025305

Offset: 0

Author

Seiichi Manyama, Feb 19 2023

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..635
• Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Crossrefs

Cf. A360592, A360747, A360699.

Programs

Mathematica

Join[{1}, Table[Sum[Binomial[n - 2*k,k] * (n - 2*k)^(2*k), {k,0,n/3}], {n,1,30}]] (* Vaclav Kotesovec, Feb 19 2023 *)

PARI

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

PARI

a(n) = sum(k=0, n\3, (n-2*k)^(2*k)*binomial(n-2*k, k));

Formula

a(n) = Sum_{k=0..floor(n/3)} (n-2*k)^(2*k) * binomial(n-2*k,k).

a(n) ~ exp(exp(4/3)*n^(1/3)/3^(1/3)) * n^(2*n/3) / 3^(2*n/3 + 1) * (1 + (3^(1/3)/(8*exp(4/3)) - 13*exp(8/3)/(6*3^(2/3))) / n^(1/3) + (67/(128*3^(1/3)*exp(8/3)) - 5*3^(2/3)*exp(4/3)/16 + 169*exp(16/3)/(216*3^(1/3))) / n^(2/3) + (3929/2304 + 497/(1024*exp(4)) + 7913*exp(4)/1728 - 2197*exp(8)/11664)/n). - Vaclav Kotesovec, Feb 19 2023