A098123 Number of compositions of n with equal number of even and odd parts.

1, 0, 0, 2, 0, 4, 6, 6, 24, 28, 60, 130, 190, 432, 770, 1386, 2856, 5056, 9828, 18918, 34908, 68132, 128502, 244090, 470646, 890628, 1709136, 3271866, 6238986, 11986288, 22925630, 43932906, 84349336, 161625288, 310404768, 596009494

Offset: 0

Author

Vladeta Jovovic, Sep 24 2004

Keywords

• easy
• nonn

Examples

			From _Gus Wiseman_, Jun 26 2022: (Start)
The a(0) = 1 through a(7) = 6 compositions (empty columns indicated by dots):
  ()  .  .  (12)  .  (14)  (1122)  (16)
            (21)     (23)  (1212)  (25)
                     (32)  (1221)  (34)
                     (41)  (2112)  (43)
                           (2121)  (52)
                           (2211)  (61)
(End)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

For partitions: A045931, ranked by A325698, strict A239241 (conj A352129).

Column k=0 of A242498.

Without multiplicity: A242821, for partitions A241638 (ranked by A325700).

These compositions are ranked by A355321.

A047993 counts balanced partitions, ranked by A106529.

A108950/A108949 count partitions with more odd/even parts.

A130780/A171966 count partitions with more or as many odd/even parts.

Cf. A000700, A000712, A001405, A026424, A027193, A028260, A097613, A240009, A277579 (ranked by A349157).

Cf. A025178.

Programs

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Count[#,?EvenQ]==Count[#,?OddQ]&]],{n,0,15}] (* Gus Wiseman, Jun 26 2022 *)

Formula

a(n) = Sum_{k=floor(n/3)..floor(n/2)} C(2*n-4*k,n-2*k)*C(n-1-k,2*n-4*k-1).

Recurrence: n*(2*n-7)*a(n) = 2*(n-2)*(2*n-5)*a(n-2) + 2*(2*n-7)*(2*n-3)*a(n-3) - (n-4)*(2*n-3)*a(n-4). - Vaclav Kotesovec, May 01 2014

a(n) ~ sqrt(c) * d^n / sqrt(Pi*n), where d = 1.94696532812840456026081823863... is the root of the equation 1-4*d-2*d^2+d^4 = 0, c = 0.225563290820392765554898545739... is the root of the equation 43*c^4-18*c^2+8*c-1=0. - Vaclav Kotesovec, May 01 2014

G.f.: 1/sqrt(1 - 4*x^3/(1-x^2)^2). - Seiichi Manyama, May 01 2025