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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.

%I A071201 #36 Jan 18 2018 17:40:50
%S A071201 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,
%T A071201 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,
%U A071201 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
%N 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.
%H A071201 Alois P. Heinz, <a href="/A071201/b071201.txt">Antidiagonals n = 1..141, flattened</a>
%H A071201 Jean-Christophe Aval, François Bergeron, <a href="http://arxiv.org/abs/1503.03991">Interlaced rectangular parking functions</a>, arXiv:1503.03991 [math.CO], 2015.
%F A071201 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).
%F A071201 Sum_{k=1..n-1} A(n-k,k) = A298072(n)-2 for n>0. - _Lee A. Newberg_, Jan 18 2018
%e A071201 Table starts:
%e A071201 1, 1, 1,  1,  1,  1, ...
%e A071201 1, 2, 2,  3,  3,  4, ...
%e A071201 1, 2, 5,  5,  7, 12, ...
%e A071201 1, 3, 5, 14, 14, 23, ...
%e A071201 1, 3, 7, 14, 42, 42, ...
%e A071201 ...
%p A071201 b:= proc(x, y, r) option remember; `if`(y<0 or y>x*r, 0,
%p A071201       `if`(x=0, 1, b(x-1, y, r) +b(x, y-1, r)))
%p A071201     end:
%p A071201 A:= (n, k)-> `if`(k<n, b(k, n, n/k), b(n, k, k/n)):
%p A071201 seq(seq(A(n, 1+d-n), n=1..d), d=1..14);  # _Alois P. Heinz_, Mar 20 2015
%t A071201 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_ *)
%Y A071201 Cf. A260419, A298072.
%K A071201 nonn,tabl
%O A071201 1,5
%A A071201 _Henry Bottomley_, May 16 2002