Solved: This exercise shows how to use generating

Chapter 7, Problem 7.4.44

(choose chapter or problem)

This exercise shows how to use generating functions to derive a formula for the sum of the first n squares. a) Show that (x2 + x)/(1 - X)4 is the generating function for the sequence {an}, where an = 12 + 2 2 + ... + n2 b) Use part (a) to find an explicit formula for the sum 1 2 + 22 + ... + n2 . The exponential generating function for the sequence {an} is the series '""" 00 a -x n n . n = O n! For example, the exponential generating function for the sequence 1, 1, 1, ... is the function L:::"= 0 x n / n! = ee. (You will find this particular series useful in these exercises.) Note that ee is the (ordinary) generating function for the sequence 1, 1, 1/2!, 1/3!, 1/4!, ....