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

2,3

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

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 *)

Crossrefs

Column k=2 of A271423.

Sequence in context: A147318 A146589 A146866 * A126363 A036836 A125997

Adjacent sequences: A271728 A271729 A271730 * A271732 A271733 A271734

Keyword

nonn

Author

Alois P. Heinz, Apr 13 2016

Status

approved