|
|
A300323
|
|
Number of Dyck paths of semilength n such that the area under the right half of the path equals the area under the left half of the path.
|
|
3
|
|
|
1, 1, 2, 3, 6, 12, 28, 69, 186, 522, 1536, 4638, 14408, 45568, 146884, 479871, 1589516, 5320854, 18000198, 61412376, 211282386, 731973720, 2553168136, 8957554412, 31604599044, 112060048354, 399227283950, 1428315878002, 5130964125124, 18499652813682
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
LINKS
|
|
|
FORMULA
|
a(n) >= A001405(n) with equality only for n <= 4.
|
|
EXAMPLE
|
/\
/ \ /\/\
a(3) = 3: / \ / \ /\/\/\ .
.
a(5) = 12 counts A001405(5) = 10 symmetric plus 2 non-symmetric Dyck paths:
/\ /\
/\/ \/ \ and its reversal.
|
|
MAPLE
|
b:= proc(x, y) option remember; expand(`if`(x=0, 1,
`if`(y<1, 0, b(x-1, y-1)*z^(2*y-1))+
`if`(x<y+2, 0, b(x-1, y+1)*z^(2*y+1))))
end:
a:= proc(n) option remember; add((p-> add(coeff(p, z, i)^2
, i=0..degree(p)))(b(n, n-2*j)), j=0..n/2)
end:
seq(a(n), n=0..32);
|
|
MATHEMATICA
|
b[x_, y_] := b[x, y] = Expand[If[x == 0, 1, If[y < 1, 0, b[x - 1, y - 1] z^(2y - 1)] + If[x < y + 2, 0, b[x - 1, y + 1] z^(2y + 1)]]];
a[n_] := a[n] = Sum[Function[p, Sum[Coefficient[p, z, i]^2, {i, 0, Exponent[p, z]}]][b[n, n - 2j]], {j, 0, n/2}];
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|