A071201 Array A(n,k) read by antidiagonals giving number of paths up and right from (0,0) to (n,k) where x/y<=n/k.

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 3, 5, 3, 1, 1, 3, 5, 5, 3, 1, 1, 4, 7, 14, 7, 4, 1, 1, 4, 12, 14, 14, 12, 4, 1, 1, 5, 12, 23, 42, 23, 12, 5, 1, 1, 5, 15, 30, 42, 42, 30, 15, 5, 1, 1, 6, 22, 55, 66, 132, 66, 55, 22, 6, 1, 1, 6, 22, 55, 99, 132, 132, 99, 55, 22, 6, 1, 1, 7, 26, 76, 143, 227, 429, 227, 143, 76, 26, 7, 1

Offset: 1

Author

Henry Bottomley, May 16 2002

Keywords

• nonn
• tabl

Examples

			Table starts:
1, 1, 1,  1,  1,  1, ...
1, 2, 2,  3,  3,  4, ...
1, 2, 5,  5,  7, 12, ...
1, 3, 5, 14, 14, 23, ...
1, 3, 7, 14, 42, 42, ...
...
		

Links

• Alois P. Heinz, Antidiagonals n = 1..141, flattened
• Jean-Christophe Aval, François Bergeron, Interlaced rectangular parking functions, arXiv:1503.03991 [math.CO], 2015.

Crossrefs

Cf. A260419, A298072.

Programs

Maple

b:= proc(x, y, r) option remember; `if`(y<0 or y>x*r, 0,
      `if`(x=0, 1, b(x-1, y, r) +b(x, y-1, r)))
    end:
A:= (n, k)-> `if`(kAlois P. Heinz, Mar 20 2015

Mathematica

b[x_, y_, r_] := b[x, y, r] = If[y < 0 || y > x*r, 0, If[x == 0, 1, b[x - 1, y, r] + b[x, y - 1, r]]]; A[n_, k_] := If[k < n, b[k, n, n/k], b[n, k, k/n]]; Table[Table[A[n, 1 + d - n], {n, 1, d}], {d, 1, 14}] // Flatten (* Jean-François Alcover, Jan 30 2016, after Alois P. Heinz *)

Formula

Some identities: A(n,k) = A(k,n); A(n,m*n) = A(n,m*n+1); A(n,n) = A000108(n); if n and k are coprime then A(n,k) = A071202(n,k).

Sum_{k=1..n-1} A(n-k,k) = A298072(n)-2 for n>0. - Lee A. Newberg, Jan 18 2018