A271731 Number of set partitions of [n] with maximal block length multiplicity equal to two.

1, 0, 9, 25, 70, 406, 2093, 10935, 41961, 267751, 1745040, 9744384, 60271016, 369277000, 2981920373, 19297914599, 136978951579, 1039245386419, 8924928983999, 65392069094065, 539711448752906, 4489189106832134, 39604974257078180, 404561197077466250

Offset: 2

Author

Alois P. Heinz, Apr 13 2016

Keywords

• nonn

Comments

At least one block length occurs exactly 2 times, and all block lengths occur at most 2 times.

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..648
• Wikipedia, Partition of a set

Crossrefs

Column k=2 of A271423.

Programs

Maple

with(combinat):
b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)
        *b(n-i*j, i-1, k)/j!, j=0..min(k, n/i))))
    end:
a:= n-> b(n$2, 2)-b(n$2, 1):
seq(a(n), n=2..30);

Mathematica

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[multinomial[n, Join[{n - i*j}, Table[i, j]]]*b[n - i*j, i - 1, k]/j!, {j, 0, Min[k, n/i] }]]];
a[n_] := b[n, n, 2] - b[n, n, 1];
Table[a[n], {n, 2, 30}] (* Jean-François Alcover, May 08 2018, after Alois P. Heinz *)