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

0,4

Links

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

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

Example

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].

Maple

g:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,
     `if`(n=0, 1, g(n, i-1)+`if`(i>n, 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 *)

Crossrefs

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

Sequence in context: A324851 A348572 A334156 * A341663 A050584 A260698

Adjacent sequences: A254998 A254999 A255000 * A255002 A255003 A255004

Keyword

nonn

Author

Alois P. Heinz, Feb 11 2015

Status

approved