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A360647 Expansion of Sum_{k>=0} (k^2 * x * (1 + x))^k.

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%I A360647 #10 Feb 16 2023 09:46:44
%S A360647 1,1,17,761,67739,10029956,2226004406,691381685259,286255287677425,
%T A360647 152360721379689043,101358756787489940837,82408168580060017122144,
%U A360647 80396790074312939684672316,92691781529853274368541343021
%N A360647 Expansion of Sum_{k>=0} (k^2 * x * (1 + x))^k.
%F A360647 a(n) = Sum_{k=0..floor(n/2)} (n-k)^(2*(n-k)) * binomial(n-k,k).
%F A360647 a(n) ~ n^(2*n). - _Vaclav Kotesovec_, Feb 16 2023
%t A360647 Flatten[{1, Table[Sum[Binomial[n-k,k] * (n-k)^(2*(n-k)), {k,0,n/2}], {n,1,30}]}] (* _Vaclav Kotesovec_, Feb 16 2023 *)
%o A360647 (PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k^2*x*(1+x))^k))
%o A360647 (PARI) a(n) = sum(k=0, n\2, (n-k)^(2*(n-k))*binomial(n-k, k));
%Y A360647 Cf. A323280, A360611.
%K A360647 nonn
%O A360647 0,3
%A A360647 _Seiichi Manyama_, Feb 15 2023