A325889 Number of colored set partitions of [2n] where colors of the elements of subsets are in (weakly) increasing order and exactly n colors are used.

1, 2, 122, 30470, 19946654, 27291293442, 67940872709600, 280154891124993313, 1787697422835498425966, 16765591042116935170071062, 221912878453525607344964295822, 4012317533096874589918210188528948, 96463460015261984561875523126569759208

Offset: 0

Author

Alois P. Heinz, Sep 07 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..150

Crossrefs

Cf. A321296.

Programs

Maple

b:= proc(n, k) option remember; `if`(n=0, 1, add(b(n-j, k)*
      binomial(n-1, j-1)*binomial(k+j-1, j), j=1..n))
    end:
a:= n-> add(b(2*n, n-i)*(-1)^i*binomial(n, i), i=0..n):
seq(a(n), n=0..15);

Mathematica

b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[b[n - j, k] Binomial[n - 1, j - 1] Binomial[k + j - 1, j], {j, 1, n}]];
a[n_] := Sum[b[2n, n - i] (-1)^i Binomial[n, i], {i, 0, n}];
a /@ Range[0, 15] (* Jean-François Alcover, Dec 15 2020, after Alois P. Heinz *)

Formula

a(n) = A321296(2n,n).