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

2,1

Links

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

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 *)

Crossrefs

Column k=2 of A321296.

Sequence in context: A228066 A009125 A037795 * A323563 A228750 A187442

Adjacent sequences: A325887 A325888 A325889 * A325891 A325892 A325893

Keyword

nonn

Author

Alois P. Heinz, Sep 07 2019

Status

approved