OFFSET
0,4
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
FORMULA
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