A045931 Number of partitions of n with equal number of even and odd parts.

1, 0, 0, 1, 0, 2, 1, 3, 2, 5, 5, 7, 9, 11, 16, 18, 25, 28, 41, 44, 62, 70, 94, 107, 140, 163, 207, 245, 302, 361, 440, 527, 632, 763, 904, 1090, 1285, 1544, 1812, 2173, 2539, 3031, 3538, 4202, 4896, 5793, 6736, 7934, 9221, 10811, 12549, 14661, 16994, 19780

Offset

0,6

Comments

The trivariate g.f. with x marking weight (i.e., sum of the parts), t marking number of odd parts and s marking number of even parts, is 1/product((1-tx^(2j-1))(1-sx^(2j)), j=1..infinity). - Emeric Deutsch, Mar 30 2006

Links

Alois P. Heinz, Table of n, a(n) for n = 0..3500 (first 1001 terms from David W. Wilson)

Formula

G.f.: Sum_{k>=0} x^(3*k)/Product_{i=1..k} (1-x^(2*i))^2. - Vladeta Jovovic, Aug 18 2007

a(n) = A000041(n)-A171967(n) = A130780(n)-A108950(n) = A171966(n)-A108949(n). - Reinhard Zumkeller, Jan 21 2010

a(n) = A000041(n) - A108950(n) - A108949(n) = A130780(n) + A171966(n) - A000041(n). - Gus Wiseman, Jan 23 2022

Example

a(9) = 5 because we have [8,1], [7,2], [6,3], [5,4] and [2,2,2,1,1,1].
From Gus Wiseman, Jan 23 2022: (Start)
The a(0) = 1 through a(12) = 9 partitions (A = 10, empty columns indicated by dots):
  ()  .  .  21   .  32   2211   43   3221   54       3322   65       4332
                    41          52   4211   63       4321   74       4431
                                61          72       4411   83       5322
                                            81       5221   92       5421
                                            222111   6211   A1       6321
                                                            322211   6411
                                                            422111   7221
                                                                     8211
                                                                     22221111
(End)

Maple

g:=1/product((1-t*x^(2*j-1))*(1-s*x^(2*j)), j=1..30): gser:=simplify(series(g, x=0, 56)): P[0]:=1: for n from 1 to 53 do P[n]:=subs(s=1/t, coeff(gser, x^n)) od: seq(coeff(t*P[n], t), n=0..53); # Emeric Deutsch, Mar 30 2006

Mathematica

p[n_] := p[n] = Select[IntegerPartitions[n], Count[#, _?OddQ] == Count[#, _?EvenQ] &]; t = Table[p[n], {n, 0, 10}] (* partitions of n with # odd parts = # even parts *)
TableForm[t] (* partitions, vertical format *)
Table[Length[p[n]], {n, 0, 30}] (* A045931 *)
(* Peter J. C. Moses, Mar 10 2014 *)

Crossrefs

The version for subsets of {1..n} is A001405.

Dominated by A027187 (partitions of even length).

More odd/even parts: A108950/A108949.

More or same number of odd/even parts: A130780/A171966.

The strict case is A239241.

This is column k = 0 of the triangle A240009.

Counting only distinct parts gives A241638, ranked by A325700.

A half-conjugate version is A277579.

These partitions are ranked by A325698.

A000041 counts integer partitions, strict A000009.

A047993 counts balanced partitions, ranked by A106529.

A257991/A257992 count odd/even parts by Heinz number.

Cf. A000070, A000700, A000712, A027193, A035363, A097613, A195017.

Sequence in context: A151533 A128100 A035579 * A325193 A079974 A102517

Adjacent sequences: A045928 A045929 A045930 * A045932 A045933 A045934

Keyword

nonn

Author

David W. Wilson

Status

approved