A086659 T(n,k) counts the set partitions of n containing k-1 blocks of length 1.

1, 1, 3, 4, 4, 6, 11, 20, 10, 10, 41, 66, 60, 20, 15, 162, 287, 231, 140, 35, 21, 715, 1296, 1148, 616, 280, 56, 28, 3425, 6435, 5832, 3444, 1386, 504, 84, 36, 17722, 34250, 32175, 19440, 8610, 2772, 840, 120, 45, 98253, 194942, 188375, 117975, 53460, 18942, 5082, 1320, 165, 55

Offset

2,3

Links

Alois P. Heinz, Rows n = 2..142, flattened

Formula

E.g.f.: exp(x*y)*(exp(exp(x)-1-x)-1). - Vladeta Jovovic, Jul 28 2003

Example

The 15 set partitions of {1,2,3,4} consist of 4 partitions with 0 blocks of length 1 : {{1,2,3,4}},{{1,2},{3,4}},{{1,3},{2,4}},{{1,4},{2,3}},
4 partitions with 1 block of length 1 : {{1},{2,3,4}},{{1,2,3},{4}},{{1,2,4},{3}},{{1,3,4},{2}}
6 partitions with 2 blocks of length 1 : {{1},{2},{3,4}},{{1},{2,3},{4}},{{1},{2,4},{3}},{{1,2},{3},{4}},{{1,3},{2},{4}},{{1,4},{2},{3}}.
(There are no partitions with n-1 blocks of length 1 and 1 with n of them)
    1;
    1,   3;
    4,   4,   6;
   11,  20,  10,  10;
   41,  66,  60,  20, 15;
  162, 287, 231, 140, 35, 21;
  ...

Maple

with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)/j!*
      b(n-i*j, i-1)*`if`(i=1, x^j, 1), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n-2))(b(n$2)):
seq(T(n), n=2..16);  # Alois P. Heinz, Mar 08 2015

Mathematica

Table[Count[Count[ #, {_Integer}]&/@SetPartitions[n], # ]&/@Range[0, n-2], {n, 2, 10}]

Crossrefs

Row sums = Bell[n]-1 (A058692), first column=A000296, main diagonal = triangular numbers A000217.

Sequence in context: A100692 A360724 A089640 * A265887 A345264 A344465

Adjacent sequences: A086656 A086657 A086658 * A086660 A086661 A086662

Keyword

nonn,tabl,easy

Author

Wouter Meeussen, Jul 27 2003

Extensions

More terms from Vladeta Jovovic, Jul 28 2003

Status

approved