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A360747 Expansion of Sum_{k>=0} (x * (1 + (k * x)^3))^k.

%I A360747 #22 Feb 20 2023 03:08:59
%S A360747 1,1,1,1,2,17,82,257,690,3484,26978,160347,726085,3529206,26885924,
%T A360747 220706533,1474182023,8834370165,65392181686,604821608674,
%U A360747 5230627589958,39543579302104,312733691925723,3013530105191283,30474809255061289
%N A360747 Expansion of Sum_{k>=0} (x * (1 + (k * x)^3))^k.
%H A360747 Vaclav Kotesovec, <a href="/A360747/b360747.txt">Table of n, a(n) for n = 0..600</a>
%H A360747 Vaclav Kotesovec, <a href="/A360747/a360747.jpg">Graph - the asymptotic ratio (30000 terms)</a>
%F A360747 a(n) = Sum_{k=0..floor(n/4)} (n-3*k)^(3*k) * binomial(n-3*k,k).
%F A360747 a(n) ~ exp(exp(9/4)*n^(1/4)/sqrt(2)) * n^(3*n/4) / 2^(3*n/2 + 2) * (1 + 1/(4*sqrt(2)*exp(9/4) * n^(1/4)) + (67/(192*exp(9/2)) - 37*exp(9/2)/16) / sqrt(n) + (497/(768*sqrt(2)*exp(27/4)) - 205*exp(9/4)/(64*sqrt(2))) / n^(3/4) + (10721/3072 + 218831/(368640*exp(9)) + (1369*exp(9))/512)/n), see graph for more minor asymptotic terms. - _Vaclav Kotesovec_, Feb 20 2023
%t A360747 Join[{1},Table[Sum[Binomial[n - 3*k,k] * (n - 3*k)^(3*k), {k,0,n/4}], {n,1,30}]] (* _Vaclav Kotesovec_, Feb 19 2023 *)
%o A360747 (PARI) my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (x*(1+(k*x)^3))^k))
%o A360747 (PARI) a(n) = sum(k=0, n\4, (n-3*k)^(3*k)*binomial(n-3*k, k));
%Y A360747 Cf. A360479, A360592, A360707.
%K A360747 nonn
%O A360747 0,5
%A A360747 _Seiichi Manyama_, Feb 19 2023