A249914 Number of partitions of 4n with equal sums of odd and even parts.

1, 1, 4, 12, 30, 70, 165, 330, 704, 1380, 2688, 4984, 9394, 16665, 29970, 52096, 90090, 152064, 257180, 423360, 697851, 1129392, 1819632, 2891520, 4583250, 7162364, 11161752, 17211180, 26427544, 40208520, 60971520, 91641748, 137290956, 204198876, 302530560

Offset

0,3

Links

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

Formula

a(n) = A000041(n) * A035294(n) = A000041(n) * A000009(2n).

a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (16*6^(3/4)*n^(7/4)). - Vaclav Kotesovec, Dec 11 2020

Example

a(0) = 1: [], the empty partition.
a(1) = 1: [2,1,1].
a(2) = 4: [4,3,1], [4,1,1,1,1], [3,2,2,1], [2,2,1,1,1,1].
a(3) = 12: [6,5,1], [6,3,3], [6,3,1,1,1], [6,1,1,1,1,1,1], [5,4,2,1], [5,2,2,2,1], [4,3,3,2], [4,3,2,1,1,1], [4,2,1,1,1,1,1,1], [3,3,2,2,2], [3,2,2,2,1,1,1], [2,2,2,1,1,1,1,1,1].

Maple

b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i<1, 0, b(n, i-2)+`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> combinat[numbpart](n) *b(2*n, 2*n-1):
seq(a(n), n=0..50);

Mathematica

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-2]+If[i>n, 0, b[n-i, i]]]];
a[n_] := PartitionsP[n] b[2n, 2n-1];
a /@ Range[0, 50] (* Jean-François Alcover, Dec 11 2020, after Alois P. Heinz *)

Crossrefs

Cf. A000009, A000041, A035294, A045931, A255001.

Sequence in context: A100691 A000298 A218009 * A006802 A068055 A221855

Adjacent sequences: A249911 A249912 A249913 * A249915 A249916 A249917

Keyword

nonn

Author

Alois P. Heinz, Feb 11 2015

Status

approved