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

0,3

Links

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

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

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 *)

Crossrefs

Sequence in context: A247978 A026592 A034275 * A151322 A351068 A345683

Adjacent sequences: A240005 A240006 A240007 * A240009 A240010 A240011

Keyword

nonn

Author

Alois P. Heinz, Mar 30 2014

Status

approved