A239261 - OEIS

%I #19 Oct 27 2023 14:42:44

%S 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 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 29970,0,0,0,52096,0,0,0,90090,0,0,0,152064,0,0,0

%N Number of partitions of n having (sum of odd parts) = (sum of even parts).

%H Alois P. Heinz, <a href="/A239261/b239261.txt">Table of n, a(n) for n = 0..500</a>

%F A239260(n) + a(n) + A239262(n) = A000041(n).

%F From _David A. Corneth_, Oct 25 2023: (Start)

%F a(4*n) = A000009(2*n) * A000041(n) for n >= 0.

%F a(4*n + r) = 0 for n >= 0 and r in {1, 2, 3}. (End)

%e a(8) counts these 4 partitions:  431, 41111, 3221, 221111.

%e From _Gus Wiseman_, Oct 24 2023: (Start)

%e The a(0) = 1 through a(12) = 12 partitions:

%e   ()  .  .  .  (211)  .  .  .  (431)     .  .  .  (633)

%e                                (3221)             (651)

%e                                (41111)            (4332)

%e                                (221111)           (5421)

%e                                                   (33222)

%e                                                   (52221)

%e                                                   (63111)

%e                                                   (432111)

%e                                                   (3222111)

%e                                                   (6111111)

%e                                                   (42111111)

%e                                                   (222111111)

%e (End)

%t z = 40; p[n_] := p[n] = IntegerPartitions[n]; f[t_] := f[t] = Length[t]

%t t1 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] < n &]], {n, z}] (* A239259 *)

%t t2 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] <= n &]], {n, z}] (* A239260 *)

%t t3 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] == n &]], {n, z}] (* A239261 *)

%t t4 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] > n &]], {n, z}] (* A239262 *)

%t t5 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] >= n &]], {n, z}] (* A239263 *)

%t (* _Peter J. C. Moses_, Mar 12 2014 *)

%Y Cf. A239259, A239260, A239262, A239263, A000041.

%Y The LHS (sum of odd parts) is counted by A113685.

%Y The RHS (sum of even parts) is counted by A113686.

%Y Without all the zeros we have a(4n) = A249914(n).

%Y The strict case (without zeros) is A255001.

%Y The Heinz numbers of these partitions are A366748, see also A019507.

%Y A000009 counts partitions into odd parts, ranks A066208.

%Y A035363 counts partitions into even parts, ranks A066207.

%Y Cf. A028260, A045931, A106529, A239241, A241638, A325698, A365067.

%K nonn,easy

%O 0,9

%A _Clark Kimberling_, Mar 13 2014

%E More terms from _Alois P. Heinz_, Mar 15 2014