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

%I #34 Feb 08 2023 15:13:45

%S 1,1,2,4,12,35,129,567,2920,16110,103467,717608,5748214,47937957,

%T 441139750,4319093093,45963368076

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

%C Differs from A331621 first at n=7.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>

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

%e a(2) = 2: 123|456, 135|246.

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

%p b:= proc(s, t) option remember; `if`(s={}, 1, (m-> add(

%p `if`({m-j, m-2*j} minus s={}, b(s minus {m, m-j, m-2*j},

%p t), 0), j=1..min(t, iquo(m-1, 2))))(max(s)))

%p end:

%p a:= proc(n) option remember; forget(b): b({$1..3*n}, n) end:

%p seq(a(n), n=0..12);

%t 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]]];

%t a[n_] := a[n] = b[Range[3n], n];

%t Table[Print[n, " ", a[n]]; a[n], {n, 0, 12}] (* _Jean-François Alcover_, May 10 2020, after Maple *)

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

%Y Main diagonal of A360334.

%K nonn,more

%O 0,3

%A _Alois P. Heinz_, Apr 20 2020