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%I A360479 #40 Feb 19 2023 10:55:25
%S A360479 1,1,1,2,9,28,81,369,1753,7323,36337,207401,1114345,6308368,40326033,
%T A360479 256982157,1658573497,11650405774,83966740913,608348063576,
%U A360479 4659734909385,36973835868521,295709600709585,2454457098977559,21106884235025305
%N A360479 Expansion of Sum_{k>=0} (x * (1 + (k * x)^2))^k.
%H A360479 Vaclav Kotesovec, <a href="/A360479/b360479.txt">Table of n, a(n) for n = 0..635</a>
%H A360479 Vaclav Kotesovec, <a href="/A360479/a360479.jpg">Graph - the asymptotic ratio (10000 terms)</a>
%F A360479 a(n) = Sum_{k=0..floor(n/3)} (n-2*k)^(2*k) * binomial(n-2*k,k).
%F A360479 a(n) ~ exp(exp(4/3)*n^(1/3)/3^(1/3)) * n^(2*n/3) / 3^(2*n/3 + 1) * (1 + (3^(1/3)/(8*exp(4/3)) - 13*exp(8/3)/(6*3^(2/3))) / n^(1/3) + (67/(128*3^(1/3)*exp(8/3)) - 5*3^(2/3)*exp(4/3)/16 + 169*exp(16/3)/(216*3^(1/3))) / n^(2/3) + (3929/2304 + 497/(1024*exp(4)) + 7913*exp(4)/1728 - 2197*exp(8)/11664)/n). - _Vaclav Kotesovec_, Feb 19 2023
%t A360479 Join[{1}, Table[Sum[Binomial[n - 2*k,k] * (n - 2*k)^(2*k), {k,0,n/3}], {n,1,30}]] (* _Vaclav Kotesovec_, Feb 19 2023 *)
%o A360479 (PARI) my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (x*(1+(k*x)^2))^k))
%o A360479 (PARI) a(n) = sum(k=0, n\3, (n-2*k)^(2*k)*binomial(n-2*k, k));
%Y A360479 Cf. A360592, A360747, A360699.
%K A360479 nonn
%O A360479 0,4
%A A360479 _Seiichi Manyama_, Feb 19 2023

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