A360699 G.f.: Sum_{k>=0} (1 + k*x)^k * x^(2*k).

1, 0, 1, 1, 1, 4, 5, 9, 28, 43, 97, 281, 507, 1286, 3666, 7494, 20470, 58725, 132484, 381700, 1113180, 2719887, 8171219, 24337511, 63524916, 197606643, 602261524, 1662206380, 5328738685, 16628469912, 48148703533, 158544768073, 506473892417, 1529218062752, 5159071807165

Offset

0,6

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Formula

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

log(a(n)) ~ n/3 * log(n/3).

a(n) ~ exp(exp(1/3)*n^(1/3)/3^(1/3)) * n^(n/3) / 3^(n/3 + 1) * (1 + (3^(1/3)/(8*exp(1/3)) - 4*exp(2/3)/3^(5/3)) / n^(1/3) + (67/(128*3^(1/3)*exp(2/3)) + 8*exp(4/3)/3^(10/3)) / n^(2/3)).

Mathematica

nmax = 40; CoefficientList[Series[Sum[(1 + k*x)^k * x^(2*k), {k, 0, nmax}], {x, 0, nmax}], x]
Join[{1}, Table[Sum[Binomial[k, n - 2*k] * k^(n - 2*k), {k, 0, n}], {n, 1, 40}]]

Crossrefs

Cf. A360592, A360707, A360479.

Sequence in context: A042715 A163868 A019148 * A336661 A352509 A222543

Adjacent sequences: A360696 A360697 A360698 * A360700 A360701 A360702

Keyword

nonn

Author

Vaclav Kotesovec, Feb 16 2023

Status

approved