A335531 Expansion of e.g.f. 1/(1-3*log(1+x)).

1, 3, 15, 114, 1152, 14562, 220842, 3907656, 79019496, 1797660000, 45439902288, 1263456328032, 38324061498672, 1259345712721392, 44565940575178992, 1689757622095909248, 68339921117338411776, 2936658673480397537664, 133615257668682429428352, 6417113656859478628233984, 324414161427519766056847104

Offset: 0

Author

Seiichi Manyama, Jun 12 2020

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..390

Crossrefs

Column k=3 of A320080.

Cf. A335530.

Programs

Mathematica

a[n_] := Sum[k! * 3^k * StirlingS1[n, k], {k, 0, n}]; Array[a, 21, 0] (* Amiram Eldar, Jun 12 2020 *)
With[{nn=20},CoefficientList[Series[1/(1-3Log[1+x]),{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Oct 02 2021 *)

PARI

a(n) = sum(k=0, n, 3^k*k!*stirling(n, k, 1));

PARI

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

PARI

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=3*sum(j=1, i, (-1)^(j-1)*(j-1)!*binomial(i, j)*v[i-j+1])); v; \\ Seiichi Manyama, May 22 2022

Formula

a(n) = Sum_{k=0..n} 3^k * k! * Stirling1(n,k).

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

a(0) = 1; a(n) = 3 * Sum_{k=1..n} (-1)^(k-1) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, May 22 2022