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%I #17 May 24 2022 12:55:33
%S 1,0,2,9,48,315,2496,22491,223728,2437371,28931040,371291283,
%T 5111412120,75014135235,1168157451384,19228202401635,333378840718944,
%U 6069073767712587,115683487658404272,2303091818149762899,47784447190060311240,1031179733234906055507
%N Expansion of e.g.f. exp( x/3 * (exp(3 * x) - 1) ).
%F a(0) = 1; a(n) = Sum_{k=2..n} k * 3^(k-2) * binomial(n-1,k-1) * a(n-k).
%F a(n) = n! * Sum_{k=0..floor(n/2)} 3^(n-2*k) * Stirling2(n-k,k)/(n-k)!.
%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(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-1, j-1)*v[i-j+1])); v;
%o (PARI) a(n) = n!*sum(k=0, n\2, 3^(n-2*k)*stirling(n-k, k, 2)/(n-k)!);
%Y Cf. A052506, A354311.
%Y Cf. A351734, A351737, A354310, A354314.
%K sign
%O 0,3
%A _Seiichi Manyama_, May 23 2022
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