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

%I #13 Feb 19 2023 09:21:05

%S 1,1,1,1,2,5,10,17,30,64,146,315,649,1386,3164,7381,16931,38649,90406,

%T 217474,527586,1277452,3112371,7705059,19336789,48800634,123617380,

%U 315671212,814711955,2119996540,5545342621,14584694613,38641783669,103158314515

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

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

%o (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, (x*(1+k*x^3))^k))

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

%Y Cf. A360592, A360748.

%Y Cf. A003269, A360731, A360747.

%K nonn

%O 0,5

%A _Seiichi Manyama_, Feb 19 2023