Chapter 8.4, Problem 54E

Generating functions are useful in studying the number of different types of partitions of an integer n. A partition of a positive integer is a way to write this integer as the sum of positive integers where repetition is allowed and the order of the integers in the sum does not matter. For example, the partitions of 5 (with no restrictions) are $1+1+1+1+1,1+1+1+2,1+1+3,$

$1+2+2,1+4,2+3$ , and 5. Exercises 53-58 illustrate some of these uses.

Show that the coefficient $p_o\left(n\right)$ of x⁷in the formal power series expansion of $1/\left(\left(1-x\right)\left(1-x^3\right)\left(1-x^5\right)-\right)$ equals the number of partitions of n into odd integers, that is, the number of ways to write n as the sum of odd positive integers, where the order does not matter and repetitions are allowed.