login
A258178
Sum over all Dyck paths of semilength n of products over all peaks p of x_p^2, where x_p is the x-coordinate of peak p.
10
1, 1, 13, 414, 24324, 2279209, 311524201, 58467947511, 14424374692879, 4525566110365523, 1759527523008436279, 830255082140922306224, 467382831980334193769718, 309419146352957449765072455, 237980526477430552734199922151, 210427994109788912088395561755374
OFFSET
0,3
COMMENTS
A Dyck path of semilength n is a (x,y)-lattice path from (0,0) to (2n,0) that does not go below the x-axis and consists of steps U=(1,1) and D=(1,-1). A peak of a Dyck path is any lattice point visited between two consecutive steps UD.
LINKS
Wikipedia, Lattice path
MAPLE
b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
`if`(x=0, 1, b(x-1, y-1, false)*`if`(t, x^2, 1) +
b(x-1, y+1, true) ))
end:
a:= n-> b(2*n, 0, false):
seq(a(n), n=0..20);
MATHEMATICA
b[x_, y_, t_] := b[x, y, t] = If[y > x || y < 0, 0, If[x == 0, 1, b[x - 1, y - 1, False]*If[t, x^2, 1] + b[x - 1, y + 1, True] ]];
a[n_] := b[2*n, 0, False];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Apr 23 2016, translated from Maple *)
KEYWORD
nonn
AUTHOR
Alois P. Heinz, May 22 2015
STATUS
approved