another partition question


how many sols we have for a + b +c + d + e = n 0<=a<=k, 0<=b<=k
same bounds for all the elements ? Had it been 1 <=a<= k i could use
the coefficient of the generating function

The problem could be changed as "number of non negative solns of n
into k parts such that each part is from a set of elements."


product of ( 1/(1-x^i)) for 1<=i<=k , the coefficient of x^n gives
the number of solns

a) Now if we have 0<=i the generating function blow up because of 1/
(1-x^0) = 1/0 , so how do we tackle this in terms of generating
functions.

b) If it had be positive partitions i could have used p(n,k) =
p(n-1,k-1 ) + p(n-k,k) but when there is 0 allowed , i get something
like p(n,k) = p(n,k-1)(add a 0 ) + p (n-1,k) (u can increase one of
the terms by 1 ). Now when increasing any of the terms by 1, we are
over computing because the element +1 needn't be in set. Any
suggestions how to remove or a better recurrence?



.

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