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%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
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