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%I #14 May 27 2022 14:36:13
%S 0,1,5,20,78,324,1520,8336,53872,405600,3492416,33798016,362543104,
%T 4264455168,54540715008,753246711808,11168972683264,176937613586432,
%U 2982069587042304,53271637651996672,1005385746384846848,19987620914387812352,417489079682758213632
%N Expansion of e.g.f. -log(1 - x) * exp(2*x).
%H Seiichi Manyama, <a href="/A346394/b346394.txt">Table of n, a(n) for n = 0..450</a>
%F a(n) = n! * Sum_{k=0..n-1} 2^k / ((n-k) * k!).
%F a(n) = Sum_{k=0..n} binomial(n,k) * A002104(k).
%F a(n) ~ exp(2) * (n-1)!. - _Vaclav Kotesovec_, Aug 09 2021
%F a(0) = 0, a(1) = 1, a(n) = (n+1) * a(n-1) - 2 * (n-1) * a(n-2) + 2^(n-1). - _Seiichi Manyama_, May 27 2022
%t nmax = 22; CoefficientList[Series[-Log[1 - x] Exp[2 x], {x, 0, nmax}], x] Range[0, nmax]!
%t Table[n! Sum[2^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+1)*v[i]-2*(i-1)*v[i-1]+2^(i-1)); v; \\ _Seiichi Manyama_, May 27 2022
%Y Cf. A002104, A010842, A066534, A346395, A346396.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jul 15 2021