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Expansion of Sum_{k>=0} (k * x * (1 + k*x^2))^k.
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%I #12 Feb 18 2023 19:00:30

%S 1,1,4,28,272,3368,50768,902397,18481408,428556075,11099001600,

%T 317544062217,9946366838784,338537433281448,12441407233436672,

%U 491002325860132371,20710640842719301632,929821866165431838038,44270378887441746923520

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

%H Winston de Greef, <a href="/A360730/b360730.txt">Table of n, a(n) for n = 0..385</a>

%F a(n) = Sum_{k=0..floor(n/3)} (n-2*k)^(n-k) * binomial(n-2*k,k).

%o (PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x*(1+k*x^2))^k))

%o (PARI) a(n) = sum(k=0, n\3, (n-2*k)^(n-k)*binomial(n-2*k, k));

%Y Cf. A360618, A360731.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Feb 18 2023