A390899 - OEIS

%I #20 Jan 20 2026 07:00:59

%S 1,6,144,5724,318168,22729248,1984198032,204688363392,24362603751744,

%T 3286210728412224,495405133989774336,82543565238795991296,

%U 15062897511253466502912,2987730767588152083973632,640021595256830429747278848,147258114060160024125094499328

%N a(n) = (1/3) * Sum_{k>=0} (2/3)^k * |Stirling1(n+k,k)|.

%H Vincenzo Librandi, <a href="/A390899/b390899.txt">Table of n, a(n) for n = 0..250</a>

%F G.f.: (1/3) * Sum_{k>=0} Product_{j=0..k-1} (2/3 + j*x).

%F a(0) = 1; a(n) = -3*a(n-1) + (9/2)*Sum_{k=0..n-1} binomial(n+1,k+1) * a(k) * a(n-1-k).

%F a(n) = (3/2)^n * A390901(n+1).

%F a(n) ~ sqrt(2) * 3^(n-1) * n^n / ((1 - log(9/4))^(n + 1/2) * exp(n)). - _Vaclav Kotesovec_, Jan 19 2026

%F E.g.f.: 1/3 - 1/(3 + 3*LambertW(-1, -3*exp(3*(-1 + x)/2)/2)). - _Vaclav Kotesovec_, Jan 20 2026

%t numTerms=18; v={1}; Do[ v=Append[v,-3 v[[-1]] +(9/2)*Sum[Binomial[n+1,k+1] v[[k+1]] v[[n-k]],{k,0,n-1}]],{n,numTerms-1}]; v (* _Vincenzo Librandi_, Jan 19 2026 *)

%t nmax = 20; Assuming[{x > 0}, CoefficientList[Series[1/3 - 1/(3 + 3*LambertW[-1, -3*E^(3*(-1 + x)/2)/2]), {x, 0, nmax}], x] * Range[0, nmax]!] (* _Vaclav Kotesovec_, Jan 20 2026 *)

%o (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-3*v[i]+9/2*sum(j=0, i-1, binomial(i+1, j+1)*v[j+1]*v[i-j])); v;

%o (Magma) N := 20; v := [1]; for n in [1..N-1] do Append(~v, -3*v[n] + (9/2)*&+[Binomial(n+1,k+1)*v[k+1]*v[n-k] : k in [0..n-1]]); end for; v; // _Vincenzo Librandi_, Jan 19 2026

%Y Cf. A390898, A390900.

%Y Cf. A390901.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Nov 23 2025