There is an identity between the divisor function of the odd numbers and the "odd" divisor function of power $3$(I don't know if there is a name for function for this type, if there is , sorry for my ignorance), that is $$\left(\sum_{n=1}^{\infty}\sigma_1(2n-1)x^{2n-1}\right)^2=\sum_{k=1}^{\infty}\left(\sum_{d|n,\frac{n}{d}\mbox{ is odd}}d^3\right)x^k$$ where $\sigma_1(n)$ is the sum of all positive divisors of $n$. This is equivalent to proving $$\left(\sum_{n=1}^{\infty}\sigma_1(2n-1)x^{2n-1}\right)^2=\sum_{k=1}^{\infty}\frac{k^3 x^k}{1-x^{2k}}$$ I tried some calculation via Mathemtica, it seems to be true.This sequence seems to be OEIS A007331, but I cannot find some way in cracking this problem. The square seems annoying and I can't get rid of it.
Thanks for your attention and helping hand.