%I #15 Aug 12 2021 11:16:52

%S 1,1,5,38,384,4854,73614,1302552,26339832,599220000,15146634096,

%T 421152109344,12774687166224,419781904240464,14855313525059664,

%U 563252540698636416,22779973705779470592,978886224493465845888,44538419222894143142784

%N Expansion of e.g.f. (1 - 2*log(1 + x))/(1 - 3*log(1 + x)).

%F a(0)=1 and a(n) = Sum_{k=0..n} k! * 3^(k-1) * Stirling1(n,k) for n > 0.

%F a(n) ~ n! * exp(1/3) / (9*(exp(1/3)-1)^(n+1)). - _Vaclav Kotesovec_, Jun 12 2020

%t a[0] = 1; a[n_] := Sum[k! * 3^(k - 1) * StirlingS1[n, k], {k, 0, n}]; Array[a, 19, 0] (* _Amiram Eldar_, Jun 12 2020 *)

%t With[{nn=20},CoefficientList[Series[(1-2Log[1+x])/(1-3Log[1+x]),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Aug 12 2021 *)

%o (PARI) {a(n) = if(n==0, 1, sum(k=0, n, k!*3^(k-1)*stirling(n, k, 1)))}

%o (PARI) N=40; x='x+O('x^N); Vec(serlaplace((1-2*log(1+x))/(1-3*log(1+x))))

%Y Column k=3 of A334369.

%Y Cf. A088501, A335531.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Jun 12 2020