This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A239261 #19 Oct 27 2023 14:42:44
%S A239261 1,0,0,0,1,0,0,0,4,0,0,0,12,0,0,0,30,0,0,0,70,0,0,0,165,0,0,0,330,0,0,
%T A239261 0,704,0,0,0,1380,0,0,0,2688,0,0,0,4984,0,0,0,9394,0,0,0,16665,0,0,0,
%U A239261 29970,0,0,0,52096,0,0,0,90090,0,0,0,152064,0,0,0
%N A239261 Number of partitions of n having (sum of odd parts) = (sum of even parts).
%H A239261 Alois P. Heinz, <a href="/A239261/b239261.txt">Table of n, a(n) for n = 0..500</a>
%F A239261 A239260(n) + a(n) + A239262(n) = A000041(n).
%F A239261 From _David A. Corneth_, Oct 25 2023: (Start)
%F A239261 a(4*n) = A000009(2*n) * A000041(n) for n >= 0.
%F A239261 a(4*n + r) = 0 for n >= 0 and r in {1, 2, 3}. (End)
%e A239261 a(8) counts these 4 partitions: 431, 41111, 3221, 221111.
%e A239261 From _Gus Wiseman_, Oct 24 2023: (Start)
%e A239261 The a(0) = 1 through a(12) = 12 partitions:
%e A239261 () . . . (211) . . . (431) . . . (633)
%e A239261 (3221) (651)
%e A239261 (41111) (4332)
%e A239261 (221111) (5421)
%e A239261 (33222)
%e A239261 (52221)
%e A239261 (63111)
%e A239261 (432111)
%e A239261 (3222111)
%e A239261 (6111111)
%e A239261 (42111111)
%e A239261 (222111111)
%e A239261 (End)
%t A239261 z = 40; p[n_] := p[n] = IntegerPartitions[n]; f[t_] := f[t] = Length[t]
%t A239261 t1 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] < n &]], {n, z}] (* A239259 *)
%t A239261 t2 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] <= n &]], {n, z}] (* A239260 *)
%t A239261 t3 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] == n &]], {n, z}] (* A239261 *)
%t A239261 t4 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] > n &]], {n, z}] (* A239262 *)
%t A239261 t5 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] >= n &]], {n, z}] (* A239263 *)
%t A239261 (* _Peter J. C. Moses_, Mar 12 2014 *)
%Y A239261 Cf. A239259, A239260, A239262, A239263, A000041.
%Y A239261 The LHS (sum of odd parts) is counted by A113685.
%Y A239261 The RHS (sum of even parts) is counted by A113686.
%Y A239261 Without all the zeros we have a(4n) = A249914(n).
%Y A239261 The strict case (without zeros) is A255001.
%Y A239261 The Heinz numbers of these partitions are A366748, see also A019507.
%Y A239261 A000009 counts partitions into odd parts, ranks A066208.
%Y A239261 A035363 counts partitions into even parts, ranks A066207.
%Y A239261 Cf. A028260, A045931, A106529, A239241, A241638, A325698, A365067.
%K A239261 nonn,easy
%O A239261 0,9
%A A239261 _Clark Kimberling_, Mar 13 2014
%E A239261 More terms from _Alois P. Heinz_, Mar 15 2014