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A113035 Number of ways the set {1,2,...,n} can be split into two subsets of which the sum of one is twice the sum of the other.

%I A113035 #19 Nov 18 2020 08:48:04
%S A113035 0,1,1,0,3,4,0,10,17,0,46,78,0,231,401,0,1233,2177,0,6869,12268,0,
%T A113035 39502,71172,0,232686,422076,0,1396669,2547246,0,8512170,15593760,0,
%U A113035 52534875,96598865,0,327669853,604405633,0,2062171364,3814087419,0,13078921499
%N A113035 Number of ways the set {1,2,...,n} can be split into two subsets of which the sum of one is twice the sum of the other.
%H A113035 Alois P. Heinz, <a href="/A113035/b113035.txt">Table of n, a(n) for n = 1..1000</a>
%F A113035 a(n) is the coefficient of x^0 in Product_{k=1..n} x^(-2k)+x^k.
%e A113035 For n=5 we have 5/1234, 14/532 and 23/541 so a(5)=3.
%p A113035 A113035:= proc(n) local i,j,p,t; t:= NULL; for j to n do p:=1; for i to j do p:=p*(x^(-2*i)+x^(i)); od; t:=t,coeff(p,x,0); od; t; end;
%p A113035 # second Maple program:
%p A113035 b:= proc(n, i) option remember; local m; m:= i*(i+1)/2;
%p A113035       `if`(n>m, 0, `if`(n=m, 1, b(abs(n-i), i-1) +b(n+i, i-1)))
%p A113035     end:
%p A113035 a:= n-> `if`(irem(n, 3)=1, 0, b(n*(n+1)/6, n)):
%p A113035 seq(a(n), n=1..60);  # _Alois P. Heinz_, Oct 31 2011
%t A113035 b[n_, i_] := b[n, i] = Module[{m = i(i+1)/2}, If[n > m, 0, If[n == m, 1, b[Abs[n - i], i - 1] + b[n + i, i - 1]]]];
%t A113035 a[n_] := If[Mod[n, 3] == 1, 0, b[n(n+1)/6, n]];
%t A113035 Array[a, 60] (* _Jean-François Alcover_, Nov 18 2020, after _Alois P. Heinz_ *)
%Y A113035 Cf. A058377, A112972.
%K A113035 nonn
%O A113035 1,5
%A A113035 _Floor van Lamoen_, Oct 11 2005