A325482 Number of colored set partitions of [n] where colors of the elements of subsets are distinct and in increasing order and exactly two colors are used.

3, 12, 41, 140, 497, 1848, 7191, 29184, 123107, 538076, 2430353, 11317644, 54229905, 266906856, 1347262319, 6965034368, 36833528195, 199037675052, 1097912385849, 6176578272780, 35409316648433, 206703355298072, 1227820993510151, 7416522514174080

Offset: 2

Author

Alois P. Heinz, Sep 06 2019

Keywords

• nonn

Examples

			a(3) = 12: 1a|2a3b, 1b|2a3b, 1a3b|2a, 1a3b|2b, 1a2b|3a, 1a2b|3b, 1a|2a|3b, 1a|2b|3a, 1b|2a|3a, 1a|2b|3b, 1b|2a|3b, 1b|2b|3a.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..792

Crossrefs

Column k=2 of A322670.

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), j=1..min(k, n)))
    end:
a:= n-> (k-> add(b(n, k-i)*(-1)^i*binomial(k, i), i=0..k))(2):
seq(a(n), n=2..27);

Mathematica

b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[b[n - j, k] Binomial[n - 1, j - 1] Binomial[k, j], {j, 1, Min[k, n]}]];
a[n_] := With[{k = 2}, Sum[b[n, k - i] (-1)^i Binomial[k, i], {i, 0, k}]];
a /@ Range[2, 27] (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz *)

Formula

E.g.f.: 1-2*exp(x)+exp(x*(x+4)/2).

a(n) ~ n^(n/2) * exp(-1 + 2*sqrt(n) - n/2) / sqrt(2). - Vaclav Kotesovec, Sep 18 2019