A360707 G.f.: Sum_{k>=0} (1 + k*x)^k * x^(3*k).

1, 0, 0, 1, 1, 0, 1, 4, 4, 1, 9, 27, 28, 16, 96, 257, 281, 250, 1251, 3161, 3665, 4321, 19489, 47685, 58662, 84099, 354739, 852216, 1110344, 1837924, 7401269, 17604002, 24221890, 44761045, 174287005, 412627144, 597640105, 1204831674, 4574415066, 10818841343

Offset

0,8

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

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

log(a(n)) ~ n/4 * log(n/4).

a(n) ~ exp(exp(1/4)*n^(1/4)/4^(1/4)) * n^(n/4) / 4^(n/4 + 1) * (1 + 1/(2^(5/2)*exp(1/4)*n^(1/4)) + (67/(192*exp(1/2)) - 15*exp(1/2)/16)/sqrt(n)).

Mathematica

nmax = 50; CoefficientList[Series[Sum[(1 + k*x)^k * x^(3*k), {k, 0, nmax}], {x, 0, nmax}], x]
Join[{1}, Table[Sum[Binomial[k, n - 3*k] * k^(n - 3*k), {k, 0, n}], {n, 1, 50}]]

Crossrefs

Cf. A360592, A360699, A360747.

Sequence in context: A123588 A289710 A243594 * A365674 A106026 A096078

Adjacent sequences: A360704 A360705 A360706 * A360708 A360709 A360710

Keyword

nonn

Author

Vaclav Kotesovec, Feb 17 2023

Status

approved