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A289474 Number of Dyck paths of semilength 4*n and height n.

%I A289474 #12 Jul 14 2017 07:06:48
%S A289474 1,1,127,26609,5828185,1244027317,258054207727,52402156308977,
%T A289474 10488803639337560,2079716852900452250,409870041643305514072,
%U A289474 80466951297324661639450,15759844503630649015980996,3082231209190859509953543989,602325527248635884295374077954
%N A289474 Number of Dyck paths of semilength 4*n and height n.
%H A289474 Alois P. Heinz, <a href="/A289474/b289474.txt">Table of n, a(n) for n = 0..400</a>
%F A289474 a(n) ~ 2^(24*n + 4) / (3^(3*n + 1/2) * 5^(5*n + 7/2) * sqrt(Pi*n)). - _Vaclav Kotesovec_, Jul 14 2017
%p A289474 b:= proc(x, y, k) option remember;
%p A289474       `if`(x=0, 1, `if`(y>0, b(x-1, y-1, k), 0)+
%p A289474       `if`(y <  min(x-1, k), b(x-1, y+1, k), 0))
%p A289474     end:
%p A289474 a:= n-> `if`(n=0, 1, b(8*n, 0, n)-b(8*n, 0, n-1)):
%p A289474 seq(a(n), n=0..20);
%t A289474 b[x_, y_, k_]:=b[x, y, k]=If[x==0, 1, If[y>0, b[x - 1, y - 1, k], 0] + If[y<Min[x - 1, k], b[x - 1, y + 1, k], 0]]; a[n_]:=a[n]=If[n==0, 1, b[8n, 0, n] - b[8n, 0, n - 1]]; Table[a[n], {n, 0, 20}] (* _Indranil Ghosh_, Jul 08 2017 *)
%Y A289474 Column k=4 of A289481.
%K A289474 nonn
%O A289474 0,3
%A A289474 _Alois P. Heinz_, Jul 06 2017