A112972 Number of ways the set {1,2,...,n} can be split into three subsets of equal sums.

0, 0, 0, 0, 1, 1, 0, 3, 9, 0, 43, 102, 0, 595, 1480, 0, 9294, 23728, 0, 157991, 411474, 0, 2849968, 7562583, 0, 53987864, 145173095, 0, 1061533318, 2885383960, 0, 21515805520, 59003023409, 0, 447142442841, 1235311936936, 0, 9489835046489, 26382363207307

Offset: 1

Author

Floor van Lamoen, Oct 07 2005

Keywords

• nonn

Examples

			For n=8 we have 84/75/6321, 84/732/651 and 831/75/642 so a(8)=3.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..125

Crossrefs

Cf. A112956, A058377.

Column k=3 of A275714.

Similar sequences: A327448, A327449, A327450.

Programs

Maple

A112972:= n-> coeff(coeff(mul((x^(-2*k)+x^k*(y^k+y^(-k)))
              , k=1..n), x, 0), y, 0)/6:
seq(A112972(n), n=1..20);
# second Maple program:
b:= proc() option remember; local i, j, t; `if`(args[1]=0,
      `if`(nargs=2, 1, b(args[t] $t=2..nargs)), add(
      `if`(args[j] -args[nargs]<0, 0, b(sort([seq(args[i]-
      `if`(i=j, args[nargs], 0), i=1..nargs-1)])[],
                args[nargs]-1)), j=1..nargs-1))
    end:
a:= n-> (m-> `if`(irem(m, 3)=0, b((m/3)$3, n)/6, 0))(n*(n+1)/2):
seq(a(n), n=1..42);  # Alois P. Heinz, Sep 03 2009

Mathematica

b[args_List] := b[args] = Module[{nargs = Length[args]}, If[args[[1]] == 0, If[nargs == 2, 1, b[args // Rest]], Sum[If[args[[j]] - Last[args] < 0, 0, b[Append[Sort[Flatten[Table[args[[i]] - If[i == j, Last[args], 0], {i, 1, nargs-1}]]], Last[args]-1]]], {j, 1, nargs-1}]]];
a[n_] := If[Mod[#, 3] == 0, b[{#/3, #/3, #/3, n}]/6, 0]&[n(n+1)/2];
Array[a, 42] (* Jean-François Alcover, Oct 30 2020, after Alois P. Heinz *)

Formula

a(n) is 1/6 of the coefficient of x^0*y^0 in Product_{k=1..n} (x^(-2*k)+x^k*(y^k+y^(-k))).

Extensions

Extended beyond a(25) by Alois P. Heinz, Sep 03 2009