A289481 Number A(n,k) of Dyck paths of semilength k*n and height n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 7, 1, 0, 1, 1, 31, 57, 1, 0, 1, 1, 127, 1341, 484, 1, 0, 1, 1, 511, 26609, 59917, 4199, 1, 0, 1, 1, 2047, 497845, 5828185, 2665884, 36938, 1, 0, 1, 1, 8191, 9096393, 517884748, 1244027317, 117939506, 328185, 1, 0

Offset

0,13

Comments

For fixed k > 1, A(n,k) ~ 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, Antidiagonals n = 0..80, flattened

Example

Square array A(n,k) begins:
  1, 1,    1,       1,          1,            1, ...
  0, 1,    1,       1,          1,            1, ...
  0, 1,    7,      31,        127,          511, ...
  0, 1,   57,    1341,      26609,       497845, ...
  0, 1,  484,   59917,    5828185,    517884748, ...
  0, 1, 4199, 2665884, 1244027317, 517500496981, ...

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, k)-> `if`(n=0, 1, b(2*n*k, 0, n)-b(2*n*k, 0, n-1)):
seq(seq(A(n, d-n), n=0..d), d=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[y<Min[x - 1, k], b[x - 1, y + 1, k], 0]]; A[n_, k_]:=A[n, k]=If[n==0, 1, b[2n*k, 0, n] - b[2n*k, 0, n - 1]]; Table[A[n, d - n], {d, 0, 12}, {n, 0, d}]//Flatten (* Indranil Ghosh, Jul 07 2017, after Maple code *)

Crossrefs

Columns k=0..10 give: A000007, A000012, A268316, A289473, A289474, A289475, A289476, A289477, A289478, A289479, A289480.

Rows n=0-2 give: A000012, A057427, A083420(k+1).

Main diagonal gives A289482.

Cf. A080936.

Sequence in context: A226371 A298937 A223855 * A229819 A194655 A197037

Adjacent sequences: A289478 A289479 A289480 * A289482 A289483 A289484

Keyword

nonn,tabl

Author

Alois P. Heinz, Jul 06 2017

Status

approved