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%I #12 May 24 2022 08:13:53
%S 1,0,2,9,60,495,4986,58401,780984,11749779,196446870,3612882933,
%T 72484364052,1575418827879,36875093680530,924769734574185,
%U 24737895033896304,703105981990977915,21159355356941587470,672148402091190649629,22475238194908656800460
%N Expansion of e.g.f. 1/(1 - x/3 * (exp(3 * x) - 1)).
%F a(0) = 1; a(n) = Sum_{k=2..n} k * 3^(k-2) * binomial(n,k) * a(n-k).
%F a(n) = n! * Sum_{k=0..floor(n/2)} 3^(n-2*k) * k! * Stirling2(n-k,k)/(n-k)!.
%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x/3*(exp(3*x)-1))))
%o (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j*3^(j-2)*binomial(i, j)*v[i-j+1])); v;
%o (PARI) a(n) = n!*sum(k=0, n\2, 3^(n-2*k)*k!*stirling(n-k, k, 2)/(n-k)!);
%Y Cf. A052848, A354313.
%Y Cf. A288834, A328182, A353999, A354312.
%K nonn
%O 0,3
%A _Seiichi Manyama_, May 23 2022