A359355 a(n) = A359107(2*n, n) = Sum_{j=0..n} Stirling2(2*n, j) = Sum_{j=0..n} A048993(2*n, j).

1, 1, 8, 122, 2795, 86472, 3403127, 164029595, 9433737120, 635182667816, 49344452550230, 4371727233798927, 437489737355466560, 49048715505983309703, 6116937802946210183545, 843220239072837883168510, 127757559136845878072576947, 21166434937698025552654090472

Offset

0,3

Comments

a(n) is the number of partitions of an 2n-set that contain at most n nonempty subsets.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..288

Formula

a(n) = A102661(2n,n) for n >= 1. - Alois P. Heinz, Jun 13 2023

a(n) ~ 2^(2*n) * n^(2*n) / (sqrt(1 + LambertW(2*n)) * LambertW(2*n)^(2*n) * exp(2*n + 1 - 2*n/LambertW(2*n))). - Vaclav Kotesovec, Jan 22 2026

Maple

b:= proc(n) option remember; `if`(n=0, 1,
      add(expand(b(n-j)*binomial(n-1, j-1)*x), j=1..n))
    end:
a:= n-> (p-> add(coeff(p, x, i), i=0..n))(b(2*n, 0)):
seq(a(n), n=0..17);  # Alois P. Heinz, Jun 13 2023

Mathematica

b[n_] := b[n] = If[n == 0, 1, Sum[Expand[b[n-j]*Binomial[n-1, j-1]*x], {j, 1, n}]];
a[n_] := With[{p = b[2*n]}, Sum[Coefficient[p, x, i], {i, 0, n}]];
Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Jul 01 2025, after Alois P. Heinz *)

Prog

(PARI) a(n) = sum(j=0, n, stirling(2*n, j, 2)); \\ Michel Marcus, Dec 27 2022

Crossrefs

Cf. A359107, A048993, A102661, A392772.

Sequence in context: A222498 A361308 A393623 * A349525 A240477 A195247

Adjacent sequences: A359352 A359353 A359354 * A359356 A359357 A359358

Keyword

nonn

Author

Peter Luschny, Dec 27 2022

Status

approved