A239261 Number of partitions of n having (sum of odd parts) = (sum of even parts).

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, 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, 29970, 0, 0, 0, 52096, 0, 0, 0, 90090, 0, 0, 0, 152064, 0, 0, 0

Offset: 0

Author

Clark Kimberling, Mar 13 2014

Keywords

• nonn
• easy

Examples

			a(8) counts these 4 partitions:  431, 41111, 3221, 221111.
From _Gus Wiseman_, Oct 24 2023: (Start)
The a(0) = 1 through a(12) = 12 partitions:
  ()  .  .  .  (211)  .  .  .  (431)     .  .  .  (633)
                               (3221)             (651)
                               (41111)            (4332)
                               (221111)           (5421)
                                                  (33222)
                                                  (52221)
                                                  (63111)
                                                  (432111)
                                                  (3222111)
                                                  (6111111)
                                                  (42111111)
                                                  (222111111)
(End)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500

Crossrefs

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

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

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

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

The strict case (without zeros) is A255001.

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

A000009 counts partitions into odd parts, ranks A066208.

A035363 counts partitions into even parts, ranks A066207.

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

Programs

Mathematica

z = 40; p[n_] := p[n] = IntegerPartitions[n]; f[t_] := f[t] = Length[t]
t1 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] < n &]], {n, z}] (* A239259 *)
t2 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] <= n &]], {n, z}] (* A239260 *)
t3 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] == n &]], {n, z}] (* A239261 *)
t4 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] > n &]], {n, z}] (* A239262 *)
t5 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] >= n &]], {n, z}] (* A239263 *)
(* Peter J. C. Moses, Mar 12 2014 *)

Formula

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

From David A. Corneth, Oct 25 2023: (Start)

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

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

Extensions

More terms from Alois P. Heinz, Mar 15 2014