A289482 Number of Dyck paths of semilength n^2 and height n.

1, 1, 7, 1341, 5828185, 517500496981, 877820839402932499, 27202373147496127842409429, 14934414860406931133627906259665137, 142143740345412121643458345045577780672138977, 23087568034858117342849941754170955046637454778184629205

Offset: 0

Author

Alois P. Heinz, Jul 06 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..40

Crossrefs

Main diagonal of A289481.

Programs

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(2*n^2, 0, n)-b(2*n^2, 0, n-1)):
seq(a(n), n=0..12);

Mathematica

b[x_, y_, k_]:=b[x, y, k]=If[x==0, 1, If[y>0, b[x - 1, y - 1, k], 0] + If[yIndranil Ghosh, Jul 08 2017 *)

Formula

a(n) = A289481(n,n).

a(n) ~ c * 2^(2*n^2) / n^4, where c = 0.034180619793706218467525729844898502557235639065782754227258170112282483988... - Vaclav Kotesovec, Jul 14 2017