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A289475 Number of Dyck paths of semilength 5*n and height n.

%I A289475 #10 Jul 14 2017 07:20:59
%S A289475 1,1,511,497845,517884748,517500496981,492342898340494,
%T A289475 450205866074148509,399858332322926629152,347831920544342751131378,
%U A289475 298149337418229660486879546,252909751775727886411508895575,212950273659595164912087912232227
%N A289475 Number of Dyck paths of semilength 5*n and height n.
%H A289475 Alois P. Heinz, <a href="/A289475/b289475.txt">Table of n, a(n) for n = 0..343</a>
%F A289475 a(n) ~ 5^(10*n + 1/2) / (2^(4*n + 3/2) * 3^(6*n + 7/2) * sqrt(Pi*n)). - _Vaclav Kotesovec_, Jul 14 2017
%p A289475 b:= proc(x, y, k) option remember;
%p A289475       `if`(x=0, 1, `if`(y>0, b(x-1, y-1, k), 0)+
%p A289475       `if`(y <  min(x-1, k), b(x-1, y+1, k), 0))
%p A289475     end:
%p A289475 a:= n-> `if`(n=0, 1, b(10*n, 0, n)-b(10*n, 0, n-1)):
%p A289475 seq(a(n), n=0..20);
%t A289475 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[10n, 0, n] - b[10n, 0, n - 1]]; Table[a[n], {n, 0, 20}] (* _Indranil Ghosh_, Jul 08 2017 *)
%Y A289475 Column k=5 of A289481.
%K A289475 nonn
%O A289475 0,3
%A A289475 _Alois P. Heinz_, Jul 06 2017