#OK to post homework
#Ariana Yousafzai, 11/15/2020, Assignment 19
1.  
c(n), which is representative of the cartesian product of AxB, may be written as the sum from k=0 to n of (n choose k)a(k)b(n-k), as k is the size of component 1, A. So, n-k must be the size of component 2, B. (n choose k) are the n labels that are included from component 1 in c(n). a(k)b(n-k) are the objects for each component from the available labels, A and B. 
If the egf of a sequence a(n) can be written as the sum from n=0 to infinity of a(n)x^n / n!, then the egs of a sequence c(n) can be written as the sum from n=0 to infinity of c(n)x^n / n! .
This is equal to the sum from n=0 to infinity of the sum of k=0 to n of a(k)x^kb(n-k)x^n-k / k!(n-k)! 
This is equal to the sum from n=0 to infinity of a(k)x^k / k! Multiplied by the sum of n=0 to infinity of b(n-k)x^n-k / (n-k)!
This is the same as egf(A) * egf(B), where C = AxB because the number of components from A included in AxB is n, while the number of components included on AxB from B is n-k. 
2.  
ogf := sum(i*(i - 1)*(i - 2)*x^i, i = 0 .. infinity)
egf:= x^3*exp(x)