A203017 Number of partitions of {1,2,...,3n} into 3 n-element subsets having the same sum.

1, 0, 1, 2, 32, 305, 4331, 63261, 1025113, 17495345, 313692810, 5838204047, 112185853894, 2213711510395, 44691175805738, 920173212324164, 19274796589413439, 409908483736507979, 8835309887111026335, 192739853119591626715, 4250191938786946069812, 94641409538083474973850

Offset: 0

Author

Alois P. Heinz, Jan 06 2012

Keywords

• nonn

Comments

The element sum of each subset is n*(3*n+1)/2 = A005449(n).

Examples

			a(0) = 1: {}, {}, {}.
a(1) = 0: there is no partition of {1,2,3} into 3 1-element subsets having the same sum.
a(2) = 1: {1,6}, {2,5}, {3,4}.
a(3) = 2: {1,5,9}, {2,6,7}, {3,4,8}; {1,6,8}, {2,4,9}, {3,5,7}.
		

Links

• Wikipedia, Partition of a set

Crossrefs

Row n=3 of A203986.

Cf. A005449, A203435.

Programs

Maple

b:= proc() option remember; local i, j, t, k, m; m:= args[nargs-1]; k:= args[nargs-0]; if args[1]=0 then `if`(nargs=3, 1, b(args[t]$t=2..nargs)) elif args[1]<1 then 0 else add(`if`(args[j] b((n*(3*n+1)/2 +n/97)$3, 3*n, n)/`if`(n>0, 6, 1):
seq(a(n), n=0..10);

Mathematica

b[args_] := b[args] = Module[{nargs = Length[args], k = args[[-1]], m = args[[-2]]}, Which[args[[1]] == 0, If[nargs == 3, 1, b[args[[2 ;; nargs]] ]], args[[1]]<1, 0, True, Sum[If[args[[j]] < m, 0, b[Join[Sort[Table[ args[[i]] - If[i == j, m + 1/97, 0], {i, 1, nargs - 2}]], {m - 1, k}]]], {j, 1, nargs-2}]]];
A[n_, k_] := If[n == 0 || k == 0, 1, b[Join[Array[(k*(n*k + 1)/2 + k/97)&, n], {k*n, k}]]/n!];
a[k_] := A[3, k];
a /@ Range[0, 10] (* Jean-François Alcover, Dec 05 2020, after Alois P. Heinz *)