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%I #16 Jul 16 2022 22:51:45
%S 0,1,5,30,256,2969,43665,776194,16159304,385353945,10353609253,
%T 309401268494,10177974023448,365446593201793,14220922741157249,
%U 596150920955286402,26783000840591098288,1283751796983110068817,65389160400251577565797
%N Expansion of e.g.f. -log(1-3*x) * exp(x)/3.
%F a(n) = n! * Sum_{k=0..n-1} 3^(n-1-k) / ((n-k) * k!).
%F a(0) = 0, a(1) = 1, a(n) = (3 * n - 2) * a(n-1) - 3 * (n-1) * a(n-2) + 1.
%F a(n) ~ (n-1)! * exp(1/3) * 3^(n-1). - _Vaclav Kotesovec_, Jun 08 2022
%o (PARI) my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-log(1-3*x)*exp(x)/3)))
%o (PARI) a(n) = n!*sum(k=0, n-1, 3^(n-1-k)/((n-k)*k!));
%o (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=0; v[2]=1; for(i=2, n, v[i+1]=(3*i-2)*v[i]-3*(i-1)*v[i-1]+1); v;
%Y Cf. A002104, A353546, A353548, A353549.
%Y Cf. A346395.
%Y Essentially partial sums of A010845.
%K nonn
%O 0,3
%A _Seiichi Manyama_, May 27 2022
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