A335531 - OEIS

%I #28 May 22 2022 09:50:16

%S 1,3,15,114,1152,14562,220842,3907656,79019496,1797660000,45439902288,

%T 1263456328032,38324061498672,1259345712721392,44565940575178992,

%U 1689757622095909248,68339921117338411776,2936658673480397537664,133615257668682429428352,6417113656859478628233984,324414161427519766056847104

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

%H Seiichi Manyama, <a href="/A335531/b335531.txt">Table of n, a(n) for n = 0..390</a>

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

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

%F 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

%t a[n_] := Sum[k! * 3^k * StirlingS1[n, k], {k, 0, n}]; Array[a, 21, 0] (* _Amiram Eldar_, Jun 12 2020 *)

%t With[{nn=20},CoefficientList[Series[1/(1-3Log[1+x]),{x,0,nn}],x] Range[ 0,nn]!] (* _Harvey P. Dale_, Oct 02 2021 *)

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

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

%o (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

%Y Column k=3 of A320080.

%Y Cf. A335530.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Jun 12 2020