Combinatorics Through Guided Discovery, 2004a

4.1. The Idea of Generating Functions 77
ones added together as the coefficient of y 0 , and in fact infinitely many numbers
added together for the coefficient of any y i . (On the other hand, it would be fine to
substitute y + y 2 for x. Can you see why?)
4.1.4 Power series
For now, most of our uses of power series will involve just simple algebra. Since
we use power series in a different way in combinatorics than we do in calculus, we
should review a bit of the algebra of power series.
• Problem 187. In the polynomial (a 0 + a 1 x + a 2 x 2 )(b 0 + b 1 x + b 2 x 2 + b 3 x 3 ),
what is the coefficient of x 2 ? What is the coefficient of x 4 ?
• Problem 188. In Problem 187 why is there a b 0 and a b 1 in your expression
for the coefficient of x 2 but there is not a b 0 or a b 1 in your expression for
the coefficient of x 4 ? What is the coefficient of x 4 in
(a 0 + a 1 x + a 2 x 2 + a 3 x 3 + a 4 x 4 )(b 0 + b 1 x + b 2 x 2 + b 3 x 3 + b 4 x 4 )?
Express this coefficient in the form
4∑
i=0
something,
where the something is an expression you need to figure out. Now suppose
that a 3 =0, a 4 =0and b 4 =0. To what is your expression equal after
you substitute these values? In particular, what does this have to do with
Problem 187? (h)
• Problem 189. The point of the Problems 187 and Problem 188 is that so long
as we are willing to assume a i =0for i > n and b j =0for j > m, then there
is a very nice formula for the coefficient of x k in the product
(
∑ n
)
a i x i
m∑
b j x j .
i=0
j=0
Write down this formula explicitly. (h)
Problem 190.
• Assuming that the rules you use to do arithmetic with polynomials
apply to power series, write down a formula for the coefficient of