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Expansion of e.g.f. exp(x^4/24 * (1+x)^4).
2

%I #13 Jun 16 2024 11:02:11

%S 1,0,0,0,1,20,180,840,1715,2520,88200,1940400,29111775,303603300,

%T 2188286100,12549537000,143029511625,3397035642000,71419225878000,

%U 1170096883956000,15075357741068625,163540869094102500,2025016641129982500,40912918773391665000

%N Expansion of e.g.f. exp(x^4/24 * (1+x)^4).

%F a(n) = n! * Sum_{k=0..floor(n/4)} binomial(4*k,n-4*k)/(24^k * k!).

%F a(0) = 1; a(n) = ((n-1)!/24) * Sum_{k=4..n} k * binomial(4,k-4) * a(n-k)/(n-k)!.

%F a(n) = (n-1)*(n-2)*(n-3)/24 * (4*a(n-4) + 20*(n-4)*a(n-5) + 36*(n-4)*(n-5)*a(n-6) + 28*(n-4)*(n-5)*(n-6)*a(n-7) + 8*(n-4)*(n-5)*(n-6)*(n-7)*a(n-8)). -_Seiichi Manyama_, Jun 16 2024

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^4/24*(1+x)^4)))

%o (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!/24*sum(j=4, i, j*binomial(4, j-4)*v[i-j+1]/(i-j)!)); v;

%Y Cf. A047974, A361567, A361568.

%Y Cf. A361280.

%K nonn

%O 0,6

%A _Seiichi Manyama_, Mar 16 2023