8. Find a generating function for the sequence (a0, a1,....ar.....), where ar=the number of non negative integral solutions to e1+e2 +....en=r where 0 less than or equal to e1 less than or equal to 1 for each i=1,2,3,.....n
The number of non-negative integral solutions to e1+e2 +...+en=r, where is equal to the number of all r-subsets of the set .
The 1-to-1 correspondence can be given as following:
The number of all r-subsets of the set , is equal to the binomial coefficient .
Therefore, the generating function for the sequence is
The final answer does not depend on , because, by definition, a generating function for the sequence (a0, a1,....ar.....) is the sum of the series , which does not depend on .
Answer.
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