A335530 Expansion of e.g.f. (1 - 2*log(1 + x))/(1 - 3*log(1 + x)).

1, 1, 5, 38, 384, 4854, 73614, 1302552, 26339832, 599220000, 15146634096, 421152109344, 12774687166224, 419781904240464, 14855313525059664, 563252540698636416, 22779973705779470592, 978886224493465845888, 44538419222894143142784

Offset: 0

Author

Seiichi Manyama, Jun 12 2020

Keywords

• nonn

Crossrefs

Column k=3 of A334369.

Cf. A088501, A335531.

Programs

Mathematica

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 *)
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 *)

PARI

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

PARI

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

Formula

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

a(n) ~ n! * exp(1/3) / (9*(exp(1/3)-1)^(n+1)). - Vaclav Kotesovec, Jun 12 2020