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A240008 Number of Dyck paths of semilength 2n such that the area between the x-axis and the path is 4n. 3
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Apr 01 2017, translated from Maple *)

CROSSREFS

Sequence in context: A247978 A026592 A034275 * A151322 A002320 A151323

Adjacent sequences:  A240005 A240006 A240007 * A240009 A240010 A240011

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Mar 30 2014

STATUS

approved

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Last modified July 7 15:31 EDT 2020. Contains 335495 sequences. (Running on oeis4.)