A255001 Number of partitions of 4n into distinct parts with equal sums of odd and even parts.

1, 0, 1, 2, 4, 6, 12, 15, 30, 40, 70, 96, 165, 216, 352, 486, 736, 988, 1518, 1998, 2944, 3952, 5607, 7488, 10614, 13916, 19305, 25536, 34854, 45568, 61864, 80240, 107640, 139776, 184832, 238680, 314628, 402800, 526176, 673652, 872592, 1110060, 1431704

Offset: 0

Author

Alois P. Heinz, Feb 11 2015

Keywords

• nonn

Examples

			a(0) = 1: [], the empty partition.
a(1) = 0.
a(2) = 1: [4,3,1].
a(3) = 2: [6,5,1], [5,4,2,1].
a(4) = 4: [8,7,1], [8,5,3], [7,6,2,1], [6,5,3,2].
		

Links

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

Crossrefs

Cf. A000009, A000700, A069910, A239241, A249914.

Programs

Maple

g:= proc(n, i) option remember; `if`(i*(i+1)/2n, 0, g(n-i, i-1))))
    end:
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-2))))
    end:
a:= n-> g(n$2)*b(2*n, 2*n-1):
seq(a(n), n=0..50);

Mathematica

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

Formula

a(n) = A000009(n) * A069910(n) = A000009(n) * A000700(2n).

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