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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.
12
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
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
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
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jul 06 2017
STATUS
approved