A287666 Number of set partitions of [n] such that j is member of block b only if b = 1 or at least one of j-1, ..., j-3 is member of a block >= b-1.

1, 1, 2, 5, 15, 52, 202, 861, 3970, 19596, 102703, 567867, 3295439, 19986462, 126231946, 827759525, 5621051650, 39439867696, 285368007479, 2125566382124, 16273261632111, 127881070062521, 1030221084660031, 8498826714433335, 71721238761675612, 618573094313147709

Offset: 0

Author

Alois P. Heinz, May 29 2017

Keywords

• nonn

Examples

			a(6) = 202 = 203 - 1 = A000110(6) - 1 counts all set partitions of [6] except: 1345|2|6.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..150
• Wikipedia, Partition of a set

Crossrefs

Column k=3 of A287641.

Cf. A000110.

Programs

Maple

b:= proc(n, l) option remember; `if`(n=0, 1, add(b(n-1,
      [seq(max(l[i], j), i=2..nops(l)), j]), j=1..l[1]+1))
    end:
a:= n-> b(n, [0$3]):
seq(a(n), n=0..26);

Mathematica

b[n_, l_] := b[n, l] = If[n == 0, 1, Sum[b[n - 1, Append[Table[Max[l[[i]], j], {i, 2, Length[l]}], j]], {j, 1, l[[1]] + 1}]];
a[n_] := b[n, {0, 0, 0}];
Table[a[n], {n, 0, 26}] (* Jean-François Alcover, May 27 2018, from Maple *)

Formula

a(n) = A287641(n,3).

a(n) = A000110(n) for n <= 5.