A360816 Expansion of Sum_{k>=0} (k*x)^(2*k) / (1 - k*x)^(k+1).

1, 0, 1, 2, 19, 100, 1118, 10034, 134993, 1715140, 27589661, 449763360, 8522965956, 168431719308, 3698624353289, 85523954588806, 2142927489388319, 56618555339223572, 1596938935380604858, 47399670488829289678, 1487559109670284821841

Offset: 0

Author

Seiichi Manyama, Feb 21 2023

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..418

Crossrefs

Cf. A072034, A360817.

Cf. A360814.

Programs

Mathematica

Join[{1}, Table[Sum[k^n * Binomial[n-k,k], {k,0,n/2}], {n,1,20}]] (* Vaclav Kotesovec, Aug 04 2025 *)

PARI

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

PARI

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

Formula

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

a(n) ~ (1-r)^(1 + n*(1-r)) * r^(1/2 + n*(1-r)) * n^n / (sqrt(1 - 2*r + 2*r^2) * (1-2*r)^(n*(1-2*r))), where r = 0.42401262950134202779147542659633991972637211375... is the root of the equation log(r*(1-r)) - 2*log(1-2*r) = 1/r. - Vaclav Kotesovec, Aug 04 2025