A289480 - OEIS

%I #11 Jul 14 2017 09:02:22

%S 1,1,524287,956185155129,2011805242484811913,

%T 3913893675608035491579363,6753921048102794214403632812402,

%U 10404372657815158859307324171401493273,14572291057533118353907127088834174993619633,18906515358804836479733610566557899759396278209535

%N Number of Dyck paths of semilength 10*n and height n.

%C In general, column k>1 of A289481 is asymptotic to 2^(2*k*n + 3) * k^(2*k*n + 1/2) / ((k-1)^((k-1)*n + 1/2) * (k+1)^((k+1)*n + 7/2) * sqrt(Pi*n)). - _Vaclav Kotesovec_, Jul 14 2017

%H Alois P. Heinz, <a href="/A289480/b289480.txt">Table of n, a(n) for n = 0..167</a>

%F a(n) ~ 2^(40*n + 7/2) * 5^(20*n + 1/2) / (3^(18*n + 1) * 11^(11*n + 7/2) * 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(20*n, 0, n)-b(20*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[20n, 0, n] - b[20n, 0, n - 1]]; Table[a[n], {n, 0, 20}] (* _Indranil Ghosh_, Jul 08 2017 *)

%Y Column k=10 of A289481.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Jul 06 2017