I try to compute the numbers in the Pascal Triangle, but on some positions (X,Y), the pascal triangle has 0, instead of the sum of P(X, Y-1) + P(X-1, Y-1) , each time when X is in relation R with Y. For example, I want P(X,Y) = 0 when R(X,y) is defined as "X is prime and Y = 2*X".
Obviously, if I am asked to compute the Pascal Triangle, I do not recursivelly compute the sums for each pair, but I compute the combinations(X,Y).
I wish to ask whether the methods of analytic combinatorics can help to solve this problem -- to find the values using some analytical formula instead of recurrence, or ad-hoc methods.
ps: EDIT TO CLARIFY
I want to compute the function c defined so:
if R(i,j): c[i][j] = 0 else: c[i][j] ← c[i-1][j-1] + c[i-1][j] and at limits as the pascal triangle.
where R is a relation of i and j, whatever R may be.
I ask if the analytic combinatorics can help, instead of using ad-hoc mathematical ideas.
As you can see, this function is almost identical with Pascal triangle (it counts the number of paths from (0,0) to (i,j) -- but I contraint it not to pass over the position (i,j)).
Can you write a generating function that can model this problem ? I never solved a problem of analytic combinatorics of this difficulty, and my only question is if somebody can help me how to write a generating function , which can be computed fast (if possible).