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%I #10 Mar 08 2021 06:17:30
%S 1,1,1,2,17,149,1439,16993,373393,6899469,214746442,7287992421
%N Number of set partitions of [3n] into n subsets of size three such that all element sums are triangular numbers.
%e a(0) = 1: {}.
%e a(1) = 1: {{1,2,3}}.
%e a(2) = 1: {{1,2,3}, {4,5,6}}.
%e a(3) = 2: {{1,6,8}, {2,4,9}, {3,5,7}}, {{1,5,9}, {2,6,7}, {3,4,8}}.
%e a(4) = 17: {{1,2,12}, {3,8,10}, {4,6,11}, {5,7,9}}, {{1,2,12}, {3,7,11}, {4,8,9}, {5,6,10}}, {{1,3,11}, {2,9,10}, {4,5,12}, {6,7,8}}, {{1,6,8}, {2,9,10}, {3,7,11}, {4,5,12}}, {{1,9,11}, {2,3,10}, {4,5,12}, {6,7,8}}, {{1,9,11}, {2,6,7}, {3,8,10}, {4,5,12}}, {{1,4,10}, {2,8,11}, {3,6,12}, {5,7,9}}, {{1,5,9}, {2,8,11}, {3,6,12}, {4,7,10}}, {{1,9,11}, {2,5,8}, {3,6,12}, {4,7,10}}, {{1,3,11}, {2,7,12}, {4,8,9}, {5,6,10}}, {{1,5,9}, {2,7,12}, {3,8,10}, {4,6,11}}, {{1,9,11}, {2,7,12}, {3,4,8}, {5,6,10}}, {{1,9,11}, {2,7,12}, {3,8,10}, {4,5,6}}, {{1,8,12}, {2,3,10}, {4,6,11}, {5,7,9}}, {{1,8,12}, {2,9,10}, {3,5,7}, {4,6,11}}, {{1,8,12}, {2,4,9}, {3,7,11}, {5,6,10}}, {{1,8,12}, {2,9,10}, {3,7,11}, {4,5,6}}.
%p b:= proc(s) option remember; `if`(s={}, 1, (j->
%p add(add(`if`(i<j and k<i and issqr((k+i+j)*8+1),
%p b(s minus {k, i, j}), 0), k=s), i=s))(max(s)))
%p end:
%p a:= n-> b({$1..3*n}):
%p seq(a(n), n=0..7);
%t b[s_] := b[s] = If[s == {}, 1, With[{j = Max[s]},
%t Sum[Sum[If[i < j && k < i && IntegerQ@Sqrt[(k + i + j)*8 + 1],
%t b[s ~Complement~ {k, i, j}], 0], {k, s}], {i, s}]]];
%t a[n_] := b[Range[3n]];
%t Table[Print[n, " ", a[n]]; a[n], {n, 0, 9}] (* _Jean-François Alcover_, Mar 08 2021, after _Alois P. Heinz_ *)
%Y Cf. A000217, A278329.
%K nonn,more
%O 0,4
%A _Alois P. Heinz_, Nov 30 2017
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