|
|
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
|
|
|
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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,5
|
|
LINKS
|
|
|
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).
|
|
EXAMPLE
|
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, ...
...
|
|
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`(k<n, b(k, n, n/k), b(n, k, k/n)):
|
|
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 *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|