A240008 Number of Dyck paths of semilength 2n such that the area between the x-axis and the path is 4n.

1, 1, 3, 14, 65, 301, 1419, 6786, 32749, 159108, 777224, 3813745, 18783934, 92811389, 459832745, 2283628771, 11364500644, 56659024320, 282939657220, 1414980598167, 7085590965083, 35523567248527, 178289298823240, 895697952270827, 4503912366189604

Offset: 0

Author

Alois P. Heinz, Mar 30 2014

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500

Programs

Maple

b:= proc(x, y, k) option remember;
      `if`(y<0 or y>x or k<0 or k>x^2/2-(y-x)^2/4, 0,
      `if`(x=0, 1, b(x-1, y-1, k-y+1/2) +b(x-1, y+1, k-y-1/2)))
    end:
a:= n-> b(4*n, 0, 4*n):
seq(a(n), n=0..30);

Mathematica

b[x_, y_, k_] := b[x, y, k] = If[y<0 || y>x || k<0 || k>x^2/2-(y-x)^2/4, 0, If[x==0, 1, b[x-1, y-1, k-y+1/2] + b[x-1, y+1, k-y-1/2]]];
a[n_] := b[4n, 0, 4n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Apr 01 2017, translated from Maple *)

Formula

a(n) = A129182(2n,4n) = A239927(4n,2n).

a(n) ~ c * d^n / sqrt(n), where d = 5.134082940807122222912767966569622... and c = 0.198313337349936555418443931967... - Vaclav Kotesovec, Apr 01 2014