What is the coefficient of $ x^{i}$ in the product $ \ \large \prod_{i \geq 1} \frac{1}{1-x^i}\prod_{i \geq 1} \frac{1}{1+x^{2i-1}}$?

Question

What is the coefficient of $ x^{i}$ in the product $ \ \large \prod_{i \geq 1} \frac{1}{1-x^i}\prod_{i \geq 1} \frac{1}{1+x^{2i-1}}$?

Answer:

$\ \large \prod_{i \geq 1} \frac{1}{1-x^i}\prod_{i \geq 1} \frac{1}{1+x^{2i-1}}$

=$\left\{(1-x)^{-1}(1-x^2)^{-1}(1-x^3)^{-1} \cdots \right\} \left\{(1+x)^{-1}(1+x^3)^{-1} (1+x^5)^{-1} \cdots \right\} $

=$ (1-x)^{-1}(1+x)^{-1}(1-x^2)^{-1}(1-x^3)^{-1}(1+x^3)^{-1} \cdots $

But I am at lost right here.

Help me to find the coefficient of $x^i$, the general coefficient.

Answer 1

The two infinite products multiplied together is the generating function of OEIS sequence A015128 which has many kinds of information about the sequence. For example,

According to Ramanujan (1913) a(n) is close to $\, (\cosh(x)-\sinh(x)/x)/(4n)$ where $\,x:=\pi\sqrt{n}.\,$

This is only anapproximation whose relative error goes to zero. If you want exact values, there are recursions such as $a(n) = -2\sum_{m=1}^{\sqrt{n}} (-1)^m a(n-m^2).$