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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.

%I A113038 #17 Jan 24 2024 10:10:43
%S A113038 0,0,0,1,0,0,5,0,0,60,0,0,747,0,0,11076,0,0,183092,0,0,3238140,0,0,
%T A113038 60475317,0,0,1175471401,0,0,23600724220,0,0,486653058995,0,0,
%U A113038 10260353188386,0,0,220439819437387,0,0,4813287355239594,0,0,106583271423691692,0,0
%N 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.
%H A113038 Alois P. Heinz, <a href="/A113038/b113038.txt">Table of n, a(n) for n = 1..100</a>
%F A113038 a(n) is half the coefficient of xy in product(x^(-2k)+x^k(y^k+y^(-k)), k=1..n) for n>1.
%e A113038 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.
%p A113038 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;
%p A113038 # second Maple program:
%p A113038 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
%t A113038 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];
%t A113038 A113038[50] (* _Jean-François Alcover_, Jan 23 2024, after first Maple program *)
%Y A113038 Cf. A112972.
%K A113038 nonn
%O A113038 1,7
%A A113038 _Floor van Lamoen_, Oct 12 2005
%E A113038 Extended beyond a(25) by _Alois P. Heinz_, Sep 03 2009