A289480 Number of Dyck paths of semilength 10*n and height n.

1, 1, 524287, 956185155129, 2011805242484811913, 3913893675608035491579363, 6753921048102794214403632812402, 10404372657815158859307324171401493273, 14572291057533118353907127088834174993619633, 18906515358804836479733610566557899759396278209535

Offset

0,3

Comments

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

Links

Alois P. Heinz, Table of n, a(n) for n = 0..167

Formula

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

Maple

b:= proc(x, y, k) option remember;
      `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))
    end:
a:= n-> `if`(n=0, 1, b(20*n, 0, n)-b(20*n, 0, n-1)):
seq(a(n), n=0..20);

Mathematica

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 *)

Crossrefs

Column k=10 of A289481.

Sequence in context: A011569 A022535 A069394 * A222530 A069280 A017702

Adjacent sequences: A289477 A289478 A289479 * A289481 A289482 A289483

Keyword

nonn

Author

Alois P. Heinz, Jul 06 2017

Status

approved