A390890 a(n) = Sum_{k>=0} (1/2)^(k+1) * Stirling2(n+k,k).

1, 2, 16, 208, 3776, 88064, 2509312, 84484096, 3281641472, 144454565888, 7106455404544, 386388342341632, 23008661588148224, 1489229440694091776, 104099447480095080448, 7815625800589565231104, 627246405151556410277888, 53587201970725717343731712, 4855511229494719127650041856

Offset

0,2

Links

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

Formula

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

E.g.f.: 1 / (2 * (1 + LambertW( -(1/2) * exp(x-1/2) ))).

a(0) = 1; a(n) = -2*n*a(n-1) + 4*Sum_{k=0..n-1} binomial(n,k+1) * a(k) * a(n-1-k).

a(n) = 2^n * A000311(n+1).

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

Mathematica

aList[N_Integer?NonNegative]:=Module[{a=ConstantArray[0, N+1]}, a[[1]]=1;
Do[a[[n+1]]=-2 n a[[n]]+4 Sum[Binomial[n, k+1] a[[k+1]] a[[n-k]], {k, 0, n-1}], {n, 1, N}]; a]
aList[18] (* Vincenzo Librandi, Jan 15 2026 *)

Prog

(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-2*i*v[i]+4*sum(j=0, i-1, binomial(i, j+1)*v[j+1]*v[i-j])); v;
(Magma) function aList(N) a := [1]; for n in [1..N] do Append(~a, -2*n*a[n] + 4*&+[Binomial(n, k+1)*a[k+1]*a[n-k] : k in [0..n-1]]); end for; return a; end function; aList(20); // Vincenzo Librandi, Jan 13 2026

Crossrefs

Cf. A390891, A390892.

Cf. A000311, A005640, A032188, A225170, A390898.

Sequence in context: A161568 A138429 A087923 * A222523 A004121 A188600

Adjacent sequences: A390887 A390888 A390889 * A390891 A390892 A390893

Keyword

nonn

Author

Seiichi Manyama, Nov 22 2025

Status

approved