Following is the wonderful Euler's partition identity: $$\prod_{i=1}^\infty (1 - x^i) = 1 + \sum_{k=1}^\infty (-1)^k \left (x^{(3k^2-k)/2} + x^{(3k^2+k)/2} \right )$$
I'm wondering if there is similar expansion for infinite product $$\prod_{i=1}^\infty (1 - x^{2i-1})$$
We know that the inverse of that is the generating function for partitions with odd parts.
Edit: After a few computation, the non-zero coefficients seem very dense and quite arbitrary, so an explicit formula might not be plausible. My main question is whether this function is $D$-finite. The notion of $D$-finite function is defined in Stanley's book. From the Euler's formula, we see that the function $$\prod (1-x^i)$$ is not $D$-finite.