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

2,1

Links

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

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

Example

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.

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 *)

Crossrefs

Column k=2 of A322670.

Sequence in context: A038345 A336337 A127120 * A017940 A038342 A260153

Adjacent sequences: A325479 A325480 A325481 * A325483 A325484 A325485

Keyword

nonn

Author

Alois P. Heinz, Sep 06 2019

Status

approved