Generating Functions (examples)
—EX1:  ak= c(n,k),  k=0,1,2,,3,…,n
—By definition,  c(n,k) =n!/(k!(n-k)!) and then c(5,2)=5!/(2!3!)=10
—G(x)=  S(0<=k<=n) c(n,k) xk  = (x+1)n           
—EX2:   ak = 1 for k=0,1,2,3,…, infinity
—G(x) = 1 + x + x2  + x3  + x4 + …  =  1/(1-x) provided that abs(x)<1
—EX3:   ak = 1/k! for k=0,1,2,3,…, infinity
—G(x) = 1/0! + x/1! + x2/2! + x3/3! + x4/4! + … = ex 
—Above equality is based upon the Taylor’s series of f(x) about x=0 which says:
— f(x) = f(0) + f ’(0) x/1! + f ’’(0) x2/2! + f ’’’(0) x3/3! 
—              + f ’’’’(0) x4 /4! + …+ f (k)(0) xk/k!  + …