cp's OEIS Frontend

A209535 Number of partitions of 0 of the form [x(1)+x(2)+...+x (j)] - [y(1)+y(2)+...+y(k)] where the x(i) are distinct positive integers <=n and the y(i) are distinct positive integers <= n.

%I A209535 #6 Mar 30 2012 18:58:15
%S A209535 1,2,6,10,19,35,60,96,160,260,404,629,953,1437,2166,3190,4634,6750,
%T A209535 9666,13762,19538,27459,38275,53159,73323,100548,137412,186696,252232,
%U A209535 339848,455448,607548,808252,1070396,1412621,1858845,2436445
%N A209535 Number of partitions of 0 of the form [x(1)+x(2)+...+x (j)] - [y(1)+y(2)+...+y(k)] where the x(i) are distinct positive integers <=n and the y(i) are distinct positive integers <= n.
%F A209535 Let q (h) be the number of partitions of h>=1 into distinct parts, as in A000009.  There are q(h)^2 ways to choose the sets {x(1),...,x(j)} and {y(1),...,y(k)} each having sum h.  Consequently, there are q(1)^2 + q(2)^2 + ... + q(n)^2 partitions of 0 as described in the Name section.
%e A209535 0 = 1-1 = 2-2 = 3-3 = 3-(1+2) = (1+2)-3 = (1+2)-(1+2),
%e A209535 so that a(3) = 6.
%t A209535 p[n_] := PartitionsQ[Range[1, n]]; l[n_] := Length[p[n]];
%t A209535 s[n_] := Apply[Plus, p[n]^2];
%t A209535 Table[s[n], {n, 1, 45}]  (* A029536 *)
%Y A209535 Cf. A209535, A000009.
%K A209535 nonn
%O A209535 1,2
%A A209535 _Clark Kimberling_, Mar 10 2012