A334250 Number of set partitions of [3n] into 3-element subsets {i, i+k, i+2k} with 1<=k<=n.

1, 1, 2, 4, 12, 35, 129, 567, 2920, 16110, 103467, 717608, 5748214, 47937957, 441139750, 4319093093, 45963368076, 510202534002, 6150655137844, 76789781005325, 1028853084775725, 14294680087131380

Offset: 0

Author

Alois P. Heinz, Apr 20 2020

Keywords

• nonn
• more

Comments

Differs from A331621 first at n=7.

Examples

			a(2) = 2: 123|456, 135|246.
a(3) = 4: 123|456|789, 123|468|579, 135|246|789, 147|258|369.
		

Links

• Wikipedia, Partition of a set

Crossrefs

Cf. A014307 (the same for 2-element subsets), A025035, A059108, A104429 (where k is not restricted), A285527, A331621, A337520.

Main diagonal of A360334.

Programs

Maple

b:= proc(s, t) option remember; `if`(s={}, 1, (m-> add(
     `if`({m-j, m-2*j} minus s={}, b(s minus {m, m-j, m-2*j},
            t), 0), j=1..min(t, iquo(m-1, 2))))(max(s)))
    end:
a:= proc(n) option remember; forget(b): b({$1..3*n}, n) end:
seq(a(n), n=0..12);

Mathematica

b[s_List, t_] := b[s, t] = If[s == {}, 1, Function[m, Sum[If[{m - j, m - 2j} ~Complement~ s == {}, b[s ~Complement~ {m, m - j, m - 2j}, t], 0], {j, 1, Min[t, Quotient[m - 1, 2]]}]][Max[s]]];
a[n_] := a[n] = b[Range[3n], n];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 12}] (* Jean-François Alcover, May 10 2020, after Maple *)

Formula

a(n) <= A104429(n) <= A025035(n).

Extensions

a(17)-a(21) from Martin Fuller, Jul 19 2025