A325890 Number of colored set partitions of [n] where colors of the elements of subsets are in (weakly) increasing order and exactly two colors are used.

3, 20, 122, 774, 5247, 38198, 298139, 2485690, 22045130, 207125874, 2053771931, 21416863948, 234145149539, 2676207794512, 31898152797430, 395584489687982, 5093960430643323, 67985187315217290, 938835976835478467, 13394336734762313862, 197153821757472332126

Offset: 2

Author

Alois P. Heinz, Sep 07 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..535

Crossrefs

Column k=2 of 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-> (k-> add(b(n, k-i)*(-1)^i*binomial(k, i), i=0..k))(2):
seq(a(n), n=2..25);

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_] := With[{k = 2}, Sum[b[n, k - i] (-1)^i Binomial[k, i], {i, 0, k}]];
a /@ Range[2, 25] (* Jean-François Alcover, Dec 15 2020, after Alois P. Heinz *)