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A255001
Number of partitions of 4n into distinct parts with equal sums of odd and even parts.
3
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
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
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Feb 11 2015
STATUS
approved