%I #19 Feb 19 2023 09:21:09

%S 1,1,4,27,257,3189,48843,889079,18730597,447945714,11983618199,

%T 354519428597,11490618543066,404910044246256,15412461332440829,

%U 630199633730994675,27548323149955792880,1282044807268698303751,63284535745130267484867

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

%F a(n) = Sum_{k=0..floor(n/4)} (n-3*k)^n * binomial(n-3*k,k).

%o (PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x*(1+(k*x)^3))^k))

%o (PARI) a(n) = sum(k=0, n\4, (n-3*k)^n*binomial(n-3*k, k));

%Y Cf. A360018, A360618.

%Y Cf. A360731.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Feb 19 2023