%I #11 Feb 20 2023 03:15:56

%S 1,0,0,1,1,0,1,4,4,1,9,27,28,16,96,257,281,250,1251,3161,3665,4321,

%T 19489,47685,58662,84099,354739,852216,1110344,1837924,7401269,

%U 17604002,24221890,44761045,174287005,412627144,597640105,1204831674,4574415066,10818841343

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

%H Seiichi Manyama, <a href="/A360707/b360707.txt">Table of n, a(n) for n = 0..1000</a>

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

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

%F 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)).

%t nmax = 50; CoefficientList[Series[Sum[(1 + k*x)^k * x^(3*k), {k, 0, nmax}], {x, 0, nmax}], x]

%t Join[{1}, Table[Sum[Binomial[k, n - 3*k] * k^(n - 3*k), {k, 0, n}], {n, 1, 50}]]

%Y Cf. A360592, A360699, A360747.

%K nonn

%O 0,8

%A _Vaclav Kotesovec_, Feb 17 2023