Various sources on the web give the result that if {ai} and {bi} satisfy $$1+\sum_{n\geq 1}^n b_n z^n = \prod_{i\geq1}\frac{1}{(1-z^i)^{a_i}}$$ then we have that $$1+B(z)=exp\sum_{k\geq1}\frac{A(z^k)}{k}$$
Setting all ai=0 then gives the generating function for the number of partitions of n, so we see that bn=p(n).
However
$$1+B(z) = \prod_{i\geq1}\frac{1}{(1-z^i)} \neq exp\sum_{k\geq1}\frac{1}{k(1-z^k)} = exp(\sum_{k\geq1}\frac{A(z^k)}{k})$$
While this equality fails, the true identity is the similar looking
$$ \prod_{i\geq1}(1-z^{-i}) = exp \sum_{k\geq1 }\frac{1}{k(1-z^k)} $$
Can anyone explain this discrepancy?