A289481 - OEIS

%I #15 May 09 2018 09:44:00

%S 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,

%T 511,26609,59917,4199,1,0,1,1,2047,497845,5828185,2665884,36938,1,0,1,

%U 1,8191,9096393,517884748,1244027317,117939506,328185,1,0

%N 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.

%C 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

%H Alois P. Heinz, <a href="/A289481/b289481.txt">Antidiagonals n = 0..80, flattened</a>

%e Square array A(n,k) begins:

%e   1, 1,    1,       1,          1,            1, ...

%e   0, 1,    1,       1,          1,            1, ...

%e   0, 1,    7,      31,        127,          511, ...

%e   0, 1,   57,    1341,      26609,       497845, ...

%e   0, 1,  484,   59917,    5828185,    517884748, ...

%e   0, 1, 4199, 2665884, 1244027317, 517500496981, ...

%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, k)-> `if`(n=0, 1, b(2*n*k, 0, n)-b(2*n*k, 0, n-1)):

%p seq(seq(A(n, d-n), n=0..d), d=0..12);

%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_, 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 *)

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

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

%Y Main diagonal gives A289482.

%Y Cf. A080936.

%K nonn,tabl

%O 0,13

%A _Alois P. Heinz_, Jul 06 2017