A113038 Number of ways the set {1,2,...,n} can be split into three subsets of which the sum of one is one more than the equal sums of both other subsets.

0, 0, 0, 1, 0, 0, 5, 0, 0, 60, 0, 0, 747, 0, 0, 11076, 0, 0, 183092, 0, 0, 3238140, 0, 0, 60475317, 0, 0, 1175471401, 0, 0, 23600724220, 0, 0, 486653058995, 0, 0, 10260353188386, 0, 0, 220439819437387, 0, 0, 4813287355239594, 0, 0, 106583271423691692, 0, 0

Offset

1,7

Links

Alois P. Heinz, Table of n, a(n) for n = 1..100

Formula

a(n) is half the coefficient of xy in product(x^(-2k)+x^k(y^k+y^(-k)), k=1..n) for n>1.

Example

For n=7 we have splittings 36/27/145, 36/127/45, 136/27/45, 135/27/46, 126/45/37 so a(7) = 5.

Maple

A113038:=proc(n) local i, j, p, t; t:= 0; for j from 2 to n do p:=1; for i to j do p:=p*(x^(-2*i)+x^i*(y^i+y^(-i))); od; t:=t, coeff(coeff(p, x, 1), y, 1)/2; od; t; end;
# 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:= proc(n) local m; m:= n*(n+1)/2; `if`(m>3 and irem(m, 3)=1, b(((m-1)/3)$2, (m-1)/3+1, n)/2, 0) end: seq(a(n), n=1..50);  # Alois P. Heinz, Sep 03 2009

Mathematica

A113038[n_] := Module[{i, j, p, t}, t = {0}; For[j = 2, j <= n, j++, p = 1; For[i = 1, i <= j, i++, p = p*(x^(-2*i) + x^i*(y^i + y^(-i))) // Expand]; t = Append[t, Coefficient[Coefficient[p, x, 1], y, 1]/2]; Print[j, " ", t[[-1]]]]; t];
A113038[50] (* Jean-François Alcover, Jan 23 2024, after first Maple program *)

Crossrefs

Cf. A112972.

Sequence in context: A221361 A083527 A221240 * A082512 A068385 A318657

Adjacent sequences: A113035 A113036 A113037 * A113039 A113040 A113041

Keyword

nonn

Author

Floor van Lamoen, Oct 12 2005

Extensions

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

Status

approved