TITLE: is this a familiar gen. fn. for partitions?
QUESTION [6 upvotes]: The $2$-adic valuation of $n\in\mathbb{N}$, denoted $\nu(n)$, is the largest power $t$ such that $2^t$ divides $n$. The number of integer partitions of $n$, denoted by $p(n)$, has generating function 
$$\sum_{n\geq0}p(n)x^n=\prod_{k\geq1}\frac1{1-x^k}.$$
However, I am finding an alternative:
$$\sum_{n\geq0}p(n)x^n=\prod_{k\geq1}(1+x^k)^{\nu(2k)}.$$

Question. Is this known? If so, any reference? If not, then any proof?

Caveat. I would be surprised if this is new.

REPLY [4 votes]: This can be proved from the famous:

Distinct parts <-> Odd parts

which can be found in Hardy & Wright : An Introduction to the Theory of Numbers.
This states:
$$(1+x)(1+x^2)(1+x^3)\dots=\frac1{(1-x)(1-x^3)(1-x^5)\dots}$$
If you substitute $x\to x^{2^k}$ for $k=1,\dots$, and multiply together, the RHS becomes the usual partition generating function, and the LHS takes your alternative form.