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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.

%I A240008 #16 Apr 02 2017 03:28:52
%S A240008 1,1,3,14,65,301,1419,6786,32749,159108,777224,3813745,18783934,
%T A240008 92811389,459832745,2283628771,11364500644,56659024320,282939657220,
%U A240008 1414980598167,7085590965083,35523567248527,178289298823240,895697952270827,4503912366189604
%N A240008 Number of Dyck paths of semilength 2n such that the area between the x-axis and the path is 4n.
%H A240008 Alois P. Heinz, <a href="/A240008/b240008.txt">Table of n, a(n) for n = 0..500</a>
%F A240008 a(n) = A129182(2n,4n) = A239927(4n,2n).
%F A240008 a(n) ~ c * d^n / sqrt(n), where d = 5.134082940807122222912767966569622... and c = 0.198313337349936555418443931967... - _Vaclav Kotesovec_, Apr 01 2014
%p A240008 b:= proc(x, y, k) option remember;
%p A240008       `if`(y<0 or y>x or k<0 or k>x^2/2-(y-x)^2/4, 0,
%p A240008       `if`(x=0, 1, b(x-1, y-1, k-y+1/2) +b(x-1, y+1, k-y-1/2)))
%p A240008     end:
%p A240008 a:= n-> b(4*n, 0, 4*n):
%p A240008 seq(a(n), n=0..30);
%t A240008 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]]];
%t A240008 a[n_] := b[4n, 0, 4n];
%t A240008 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Apr 01 2017, translated from Maple *)
%K A240008 nonn
%O A240008 0,3
%A A240008 _Alois P. Heinz_, Mar 30 2014