A360747 Expansion of Sum_{k>=0} (x * (1 + (k * x)^3))^k.

1, 1, 1, 1, 2, 17, 82, 257, 690, 3484, 26978, 160347, 726085, 3529206, 26885924, 220706533, 1474182023, 8834370165, 65392181686, 604821608674, 5230627589958, 39543579302104, 312733691925723, 3013530105191283, 30474809255061289

Offset: 0

Author

Seiichi Manyama, Feb 19 2023

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..600
• Vaclav Kotesovec, Graph - the asymptotic ratio (30000 terms)

Crossrefs

Cf. A360479, A360592, A360707.

Programs

Mathematica

Join[{1},Table[Sum[Binomial[n - 3*k,k] * (n - 3*k)^(3*k), {k,0,n/4}], {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)^3))^k))

PARI

a(n) = sum(k=0, n\4, (n-3*k)^(3*k)*binomial(n-3*k, k));

Formula

a(n) = Sum_{k=0..floor(n/4)} (n-3*k)^(3*k) * binomial(n-3*k,k).

a(n) ~ exp(exp(9/4)*n^(1/4)/sqrt(2)) * n^(3*n/4) / 2^(3*n/2 + 2) * (1 + 1/(4*sqrt(2)*exp(9/4) * n^(1/4)) + (67/(192*exp(9/2)) - 37*exp(9/2)/16) / sqrt(n) + (497/(768*sqrt(2)*exp(27/4)) - 205*exp(9/4)/(64*sqrt(2))) / n^(3/4) + (10721/3072 + 218831/(368640*exp(9)) + (1369*exp(9))/512)/n), see graph for more minor asymptotic terms. - Vaclav Kotesovec, Feb 20 2023