%I #10 Jul 14 2017 08:36:43
%S 1,1,8191,164531565,3673214880049,77462600751077244,
%T 1505240258416480353423,27202373147496127842409429,
%U 464106749942563876038980247765,7576947003340172511554825394061140,119634586370431286462528705183632896422
%N Number of Dyck paths of semilength 7*n and height n.
%H Alois P. Heinz, <a href="/A289477/b289477.txt">Table of n, a(n) for n = 0..241</a>
%F a(n) ~ 7^(14*n + 1/2) / (2^(16*n + 8) * 3^(6*n + 1/2) * sqrt(Pi*n)). - _Vaclav Kotesovec_, Jul 14 2017
%p b:= proc(x, y, k) option remember;
%p `if`(x=0, 1, `if`(y>0, b(x-1, y-1, k), 0)+
%p `if`(y < min(x-1, k), b(x-1, y+1, k), 0))
%p end:
%p a:= n-> `if`(n=0, 1, b(14*n, 0, n)-b(14*n, 0, n-1)):
%p seq(a(n), n=0..20);
%t 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[14n, 0, n] - b[14n, 0, n - 1]]; Table[a[n], {n, 0, 20}] (* _Indranil Ghosh_, Jul 07 2017, after Maple code *)
%Y Column k=7 of A289481.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Jul 06 2017