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%I #14 May 27 2022 14:36:17
%S 0,1,7,38,192,969,5115,29322,187992,1370745,11392839,107043606,
%T 1122823944,12989320785,164040593067,2243143392138,32994768719376,
%U 519229765892241,8701862242296807,154700700117472422,2907409255935736752,57588370882960384377,1198954118077558162875
%N Expanison of e.g.f. -log(1 - x) * exp(3*x).
%H Seiichi Manyama, <a href="/A346395/b346395.txt">Table of n, a(n) for n = 0..450</a>
%F a(n) = n! * Sum_{k=0..n-1} 3^k / ((n-k) * k!).
%F a(n) ~ exp(3) * (n-1)!. - _Vaclav Kotesovec_, Aug 09 2021
%F a(0) = 0, a(1) = 1, a(n) = (n+2) * a(n-1) - 3 * (n-1) * a(n-2) + 3^(n-1). - _Seiichi Manyama_, May 27 2022
%t nmax = 22; CoefficientList[Series[-Log[1 - x] Exp[3 x], {x, 0, nmax}], x] Range[0, nmax]!
%t Table[n! Sum[3^k/((n - k) k!), {k, 0, n - 1}], {n, 0, 22}]
%o (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=0; v[2]=1; for(i=2, n, v[i+1]=(i+2)*v[i]-3*(i-1)*v[i-1]+3^(i-1)); v; \\ _Seiichi Manyama_, May 27 2022
%Y Cf. A002104, A053486, A346394, A346396.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jul 15 2021
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