A390892 a(n) = (1/4) * Sum_{k>=0} (3/4)^k * Stirling2(n+k,k).

1, 12, 480, 31872, 2961408, 353722368, 51635159040, 8907631362048, 1773032473362432, 399965218010038272, 100838862518338191360, 28099340559309216940032, 8575717805047407704014848, 2844822675521564663001120768, 1019206701303290275069396254720

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..280

Formula

G.f.: (1/4) * Sum_{k>=0} (3/4)^k / Product_{j=1..k} (1 - j*x).

E.g.f.: 1 / (4 * (1 + LambertW( -(3/4) * exp(x-3/4) ))).

a(0) = 1; a(n) = -4*n*a(n-1) + 16*Sum_{k=0..n-1} binomial(n,k+1) * a(k) * a(n-1-k).

a(n) = 4^n * A201466(n+1).

a(n) ~ 2^(2*n-1) * n^n / ((4*log(4/3) - 1)^(n + 1/2) * exp(n)). - Vaclav Kotesovec, Jan 20 2026

Mathematica

a[n_]:=Module[{v={1}}, Do[v=Join[v, {-4 i v[[i]]+16 Sum[Binomial[i, k+1] v[[k+1]] v[[i-k]], {k, 0, i-1}]}], {i, 1, n}]; v]; a[15] (* Vincenzo Librandi, Jan 05 2026 *)

Prog

(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-4*i*v[i]+16*sum(j=0, i-1, binomial(i, j+1)*v[j+1]*v[i-j])); v;
(Magma) n := 20; v := [1] cat [0 : i in [1..n]]; for i in [1..n] do v[i+1] := -4*i*v[i] + 16*&+[ Binomial(i, k+1)*v[k+1]*v[i-k] : k in [0..i-1] ]; end for; v; // Vincenzo Librandi, Jan 05 2026

Crossrefs

Cf. A390890, A390891.

Cf. A201466, A234294.

Sequence in context: A012460 A012685 A012465 * A012394 A012686 A012466

Adjacent sequences: A390889 A390890 A390891 * A390893 A390894 A390895

Keyword

nonn

Author

Seiichi Manyama, Nov 22 2025

Status

approved