[This is a partner question to http://codegolf.stackexchange.com/questions/51406/calculate-a-probability-exactly ]

This task is about writing code to compute a probability exactly and quickly. The output should be a precise probability written as a fraction in its most reduced form. That is it should never output 4/8 but rather 1/2.

For some positive integer n, consider a uniformly random string of 1s and -1s of length n and call it A. Now concatenate to A its first value. That is A[1] = A[n+1] if indexing from 1. A now has length n+1. Now also consider a second random string of length n whose first n values are -1, 0, or 1 with probability 1/4,1/2, 1/4 each and call it B.

For example, consider n=3. Possible values for A and B could be A = [-1,1,1,-1] and B=[0,1,-1]. In this case the two inner products are 0 and 2.

Now consider the inner product of A[1,...,n] and B and the inner product of A[2,...,n+1] and B.

Your code must output the probability that both inner products are zero.

Copying the table produced by Martin Büttner we have the following sample results.

n P(n) 1 1/2 2 3/8 3 7/32 4 89/512 5 269/2048 6 903/8192 7 3035/32768 8 169801/2097152 

Languages and libraries

You can use any freely available language and libraries you like. I must be able to run your code so please include a full explanation for how to run/compile your code in linux if at all possible.

The task

Your code must start with n=1 and give the correct output for each increasing n on a separate line. It should stop after 10 seconds.

The score

The score is simply the highest n reached before your code stops after 10 seconds when run on my computer. If there is a tie, the winner be the one to get to the highest score quickest.