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Expansion of e.g.f. 1/(1 - x^3/6 * (exp(x) - 1)).
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%I #10 May 13 2022 12:05:58

%S 1,0,0,0,4,10,20,35,1176,10164,58920,277365,3363580,47567806,

%T 519759604,4591587455,51017687280,786120055400,12187597925136,

%U 165128862881769,2261843835692340,36940778814100210,678763188831800380,12143893591131411571,211404290379223149384

%N Expansion of e.g.f. 1/(1 - x^3/6 * (exp(x) - 1)).

%F a(0) = 1; a(n) = n!/6 * Sum_{k=4..n} 1/(k-3)! * a(n-k)/(n-k)! = binomial(n,3) * Sum_{k=4..n} binomial(n-3,k-3) * a(n-k).

%F a(n) = n! * Sum_{k=0..floor(n/4)} k! * Stirling2(n-3*k,k)/(6^k * (n-3*k)!).

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x^3/6*(exp(x)-1))))

%o (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i!/6*sum(j=4, i, 1/(j-3)!*v[i-j+1]/(i-j)!)); v;

%o (PARI) a(n) = n!*sum(k=0, n\4, k!*stirling(n-3*k, k, 2)/(6^k*(n-3*k)!));

%Y Cf. A052848, A353998.

%Y Cf. A000292, A351506, A354001.

%K nonn

%O 0,5

%A _Seiichi Manyama_, May 13 2022