A360748 Expansion of Sum_{k>=0} (x * (1 + k*x^2))^k.

1, 1, 1, 2, 5, 10, 21, 53, 133, 327, 861, 2361, 6469, 18168, 52757, 155221, 463077, 1412656, 4379917, 13747504, 43834213, 141866555, 464650309, 1541008295, 5176660997, 17586913779, 60400627453, 209746820056, 735953607173, 2607716976945, 9330605338485

Offset: 0

Author

Seiichi Manyama, Feb 19 2023

Keywords

• nonn

Links

• Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Crossrefs

Cf. A360592, A360749.

Cf. A000930, A360730, A360479.

Programs

Mathematica

Join[{1},Table[Sum[Binomial[n - 2*k,k] * (n - 2*k)^k, {k,0,n/3}], {n,1,30}]] (* Vaclav Kotesovec, Feb 20 2023 *)

PARI

my(N=40, 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)^k*binomial(n-2*k, k));

Formula

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

a(n) ~ exp(exp(2/3)*n^(2/3)/3^(2/3) - 5*exp(4/3)*n^(1/3)/(18*3^(1/3)) + 22*exp(2)/81) * n^(n/3) / 3^(n/3 + 1) * (1 + (2*exp(2/3)/3^(5/3) - 3295*exp(8/3)/(2916*3^(2/3)))/n^(1/3) + (3^(2/3)/(8*exp(2/3)) + 35*exp(4/3)/(36*3^(1/3)) + 27379*exp(10/3)/(17496*3^(1/3)) + 10857025*exp(16/3)/(51018336*3^(1/3)))/n^(2/3)). - Vaclav Kotesovec, Feb 20 2023