I'm reading Analytic Combinatorics [PDF] book by Flajolet and Sedgewick, and I can't figure out one of the steps in the derivation of the $P_n$ — number of partitions of size $n$ (or coefficients in Taylor expansion of $P(z)$, see below).
On page 42, I.13. says:
$\rhd$ I.13. A recurrence for the partition numbers. Logarithmic differentiation gives z \frac{P\,'(z)}{P(z)} = \sum_{n=1}^\infty \frac{n z^n}{1-z^n} \qquad \text{implying} \qquad n P_n = \sum_{j=1}^n \, \sigma(j) P_{n-j}\,, where $\sigma(n)$ is the sum of divisors of $n$ (e.g., $\sigma(6)=1+2+3+6=12$).
$P(z)$ is earlier defined ($(39)$ on page 41) as $P(z) = \prod_{n=1}^\infty \frac{1}{1-z^n}.$
I understand how they get the part to the left of the word “implying”, however I fail to see how they get the part to the right of it. How do they get it?