[Math] partitioning 10 into odd parts using generating function.

Question

Use generating functions to find the number of ways to partition 10 into odd parts.

I am not really sure how to go about doing this. i know it is a product series, but i am not sure how to find the partitions

Answer 1

Hint: The ordinary generating function for odd partitions is $$ A(x)=\frac{x}{1-x^2}. $$ Why? If $a_n$ is the number of ways that you can partition $n$ in to a single odd number, then $a_n=1$ if $n$ is odd (you can write the number as itself) and $a_n=0$ if $n$ is even; thus $$ A(x)=\sum_{n=0}^{\infty}a_nx^n=\sum_{k=0}^{\infty}x^{2k+1}=x\sum_{k=0}^{\infty}x^{2k}=x\cdot\frac{1}{1-x^2},\qquad\lvert x\rvert<1. $$ Now, what does the coefficient by $x^n$ in $A(x)\cdot A(x)$ tell you? How about in $(A(x))^3$? Or for $(A(x))^k$ in general?