Question 1
What is the coefficient of x50y50 in the bi-Taylor expansion (around (0,0)) of the rational function
1/(1-a[1]*x-a[2]*y+a[5]*x*y-a[6]*x^3+a[7]*y^3)

Answer 1
I tried expanding it to 50 terms but maple is not happy about it. Maybe I am doing something wrong in that regard. 
This is the code I tried to send in
DiagSeq2(1/(-8*x^3 + 6*y^3 - x + xy - 8*y + 1), x, y, 3)

 In how many ways can you walk from [0,0,0] to [30,30,30] if the fundamental ("atomic") steps are
{[a[1],a[3],a[9]], [a[2], a[4], a[5], [a[4],a[3],a[3]}

DiagWalks3D({[1, 4, 4], [1, 4, 8], [2, 1, 1]}, 30);
[1, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 28, 0, 0, 0, 0, 0, 0, 220, 0, 0, 0, 0, 0, 0, 1820, 0, 0]

Since the last index is 0 you cannot get to (30, 30, 30) using these atomic steps. 

Question 2
Write a Maple procedure WordsMod(a,S,x) that inputs a positive integer a, subsest S of {1,2,..., a} and a variable x and outputs the rational function that is the generating function of finite sequence of positive integers that leave remainder that belongs to S when divided by a. Use the Maple "factor" to make it as nice as possible.

Answer 2:

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