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Expansion of e.g.f. exp(x^4/(24 * (1 - x)^3)).
1

%I #10 Jun 17 2024 08:50:24

%S 1,0,0,0,1,15,180,2100,25235,319410,4299750,61815600,950524575,

%T 15633092475,274749725250,5151569172750,102831791687625,

%U 2179782464359500,48933251188321500,1160002995644493000,28956069155772383625,759014081927743516875

%N Expansion of e.g.f. exp(x^4/(24 * (1 - x)^3)).

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

%F a(0) = 1; a(n) = ((n-1)!/24) * Sum_{k=4..n} k * binomial(k-2,k-4) * a(n-k)/(n-k)!.

%o (PARI) a(n) = n!*sum(k=0, n\4, binomial(n-k-1, n-4*k)/(24^k*k!));

%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(j-2, j-4)*v[i-j+1]/(i-j)!)); v;

%Y Cf. A361545, A361577, A373758.

%K nonn

%O 0,6

%A _Seiichi Manyama_, Jun 17 2024