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A360611 Expansion of Sum_{k>=0} (k * x * (1 + x))^k.

%I A360611 #14 Feb 18 2023 22:48:57
%S A360611 1,1,5,35,341,4230,63844,1135753,23273363,539881365,13986073419,
%T A360611 400227436252,12538263892232,426810214125441,15687071552060221,
%U A360611 619144491880324087,26117514728711229877,1172635546310430028562,55833864788507320490268
%N A360611 Expansion of Sum_{k>=0} (k * x * (1 + x))^k.
%H A360611 Winston de Greef, <a href="/A360611/b360611.txt">Table of n, a(n) for n = 0..385</a>
%F A360611 a(n) = Sum_{k=0..floor(n/2)} (n-k)^(n-k) * binomial(n-k,k).
%F A360611 a(n) ~ exp(exp(-1)) * n^n. - _Vaclav Kotesovec_, Feb 14 2023
%t A360611 Flatten[{1, Table[Sum[Binomial[n-k, k] * (n-k)^(n-k), {k, 0, n/2}], {n, 1, 20}]}] (* _Vaclav Kotesovec_, Feb 14 2023 *)
%o A360611 (PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=0,N, (k*x*(1+x))^k))
%o A360611 (PARI) a(n) = sum(k=0,n\2, (n-k)^(n-k)*binomial(n-k, k));
%Y A360611 Cf. A355494, A360592.
%K A360611 nonn
%O A360611 0,3
%A A360611 _Seiichi Manyama_, Feb 14 2023