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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. 6

%I

%S 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 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 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 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 Alois P. Heinz, <a href="/A071201/b071201.txt">Antidiagonals n = 1..141, flattened</a>

%H Jean-Christophe Aval, Fran├žois Bergeron, <a href="http://arxiv.org/abs/1503.03991">Interlaced rectangular parking functions</a>, arXiv:1503.03991 [math.CO], 2015.

%F 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 Sum_{k=1..n-1} A(n-k,k) = A298072(n)-2 for n>0. - _Lee A. Newberg_, Jan 18 2018

%e Table starts:

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

%e 1, 2, 2, 3, 3, 4, ...

%e 1, 2, 5, 5, 7, 12, ...

%e 1, 3, 5, 14, 14, 23, ...

%e 1, 3, 7, 14, 42, 42, ...

%e ...

%p b:= proc(x, y, r) option remember; `if`(y<0 or y>x*r, 0,

%p `if`(x=0, 1, b(x-1, y, r) +b(x, y-1, r)))

%p end:

%p A:= (n, k)-> `if`(k<n, b(k, n, n/k), b(n, k, k/n)):

%p seq(seq(A(n, 1+d-n), n=1..d), d=1..14); # _Alois P. Heinz_, Mar 20 2015

%t 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-Fran├žois Alcover_, Jan 30 2016, after _Alois P. Heinz_ *)

%Y Cf. A260419, A298072.

%K nonn,tabl

%O 1,5

%A _Henry Bottomley_, May 16 2002

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Last modified February 22 18:05 EST 2019. Contains 320400 sequences. (Running on oeis4.)