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

0,4

Links

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

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

Example

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)

Mathematica

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

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).

Sequence in context: A192134 A322402 A196877 * A066659 A343468 A287846

Adjacent sequences: A098120 A098121 A098122 * A098124 A098125 A098126

Keyword

easy,nonn

Author

Vladeta Jovovic, Sep 24 2004

Status

approved