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
Keywords
Examples
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.
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..100
Crossrefs
Cf. A112972.
Programs
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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
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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 *)
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.
Extensions
Extended beyond a(25) by Alois P. Heinz, Sep 03 2009