|P is for Practical|
On a related note; a real challenge is also to just compute one of the entries in the table. Especially for higher values.
If we number the table as follows:
it can be said to define a function F(n,m). For instance:
However for higher values of n this function returns very large numbers around the point F(n,n/2). This are the numbers in the center of the table. For instance try computing F(400,200) or F(400,199).
These numbers are so large they do not fit into a floatingpoint number. It's of course possible to use the BigInt package, but that slows down computation, and uses quite some memory. Hence just printing the table for large values of n becomes impossible.
However the numbers around F(n,n), and F(n,1) stay pretty small for large n and could be computed. These are the numbers at the edge of the table. For instance F(400,1) = F(400,400) = 1, and even F(400,5) is computable.
Finding F(n,m) is useful for several statistical and counting problems. However writing a program that efficiently computes F(n,m) for any n,m, is quite a challenge! (At least I found it quite a challenge when I tried to write one.) One approach is to 'walk' the edge of the table towards the entry that needs to be computed. Thereby avoiding the center of the table with all the giant numbers. But there must be a more effecient way to compute F .... any suggestions?