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A249914 Number of partitions of 4n with equal sums of odd and even parts. 2
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified May 16 19:46 EDT 2021. Contains 343951 sequences. (Running on oeis4.)