How to prove this infinite product identity?

How can I prove the following identity?$$\large\prod_{k=1}^\infty\frac1{1-2^{1-2k}}=\sum_{m=0}^\infty\left(2^{-\frac{m^2+m}{2}}\prod_{n=1}^\infty\frac{1-2^{-m-n}}{1-2^{-n}}\right)$$Numerically both sides evaluate to$$2.38423102903137172414989928867839723877...$$