A324523 Number of colored set partitions of [2n] where elements of subsets have distinct colors and exactly n colors are used.

1, 1, 74, 31770, 42687960, 134092967400, 831428629796160, 9095459029214397840, 162061482211484681105280, 4429476877635332233622271360, 177245727799376537644530489120000, 10002691163041098923871227379695673600, 772102922309973700712743861257373871078400

Offset: 0

Author

Alois P. Heinz, Sep 03 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..139
• Wikipedia, Partition of a set

Crossrefs

Cf. A323128.

Programs

Maple

b:= proc(n, k) option remember; `if`(n=0, 1, add(k!/(k-j)!
      *binomial(n-1, j-1)*b(n-j, k), j=1..min(k, 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[k!/(k-j)! Binomial[n - 1, j - 1]* b[n - j, k], {j, 1, Min[k, n]}]];
a[n_] := Sum[b[2n, n - i] (-1)^i Binomial[n, i], {i, 0, n}];
a /@ Range[0, 15] (* Jean-François Alcover, May 05 2020, after Maple *)

Formula

a(n) = A323128(2n,n).