A390899 a(n) = (1/3) * Sum_{k>=0} (2/3)^k * |Stirling1(n+k,k)|.

1, 6, 144, 5724, 318168, 22729248, 1984198032, 204688363392, 24362603751744, 3286210728412224, 495405133989774336, 82543565238795991296, 15062897511253466502912, 2987730767588152083973632, 640021595256830429747278848, 147258114060160024125094499328

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..250

Formula

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

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).

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

a(n) ~ sqrt(2) * 3^(n-1) * n^n / ((1 - log(9/4))^(n + 1/2) * exp(n)). - Vaclav Kotesovec, Jan 19 2026

E.g.f.: 1/3 - 1/(3 + 3*LambertW(-1, -3*exp(3*(-1 + x)/2)/2)). - Vaclav Kotesovec, Jan 20 2026

Mathematica

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 *)
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 *)

Prog

(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;
(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

Crossrefs

Cf. A390898, A390900.

Cf. A390901.

Sequence in context: A208230 A010043 A270504 * A085905 A203978 A090443

Adjacent sequences: A390896 A390897 A390898 * A390900 A390901 A390902

Keyword

nonn

Author

Seiichi Manyama, Nov 23 2025

Status

approved