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

0,4

Comments

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

Links

Table of n, a(n) for n=0..21.

Wikipedia, Partition of a set

Example

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}.

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]<m, 0, b(sort([seq(args[i] -`if`(i=j, m+1/97, 0), i=1..nargs-2)])[], m-1, k)), j=1..nargs-2) fi end:
a:= n-> 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 *)

Crossrefs

Row n=3 of A203986.

Cf. A005449, A203435.

Sequence in context: A158040 A202746 A212797 * A079766 A224293 A053065

Adjacent sequences: A203014 A203015 A203016 * A203018 A203019 A203020

Keyword

nonn

Author

Alois P. Heinz, Jan 06 2012

Status

approved