A q-series related to adjoint representation of lie group

Question

What is the infinite sum expansion in the degree of q?

$$ \exp \left[\sum_{n=1}^\infty \frac{1}{n}\frac{2q^n }{1-q^n } \right] = \prod_{n=1}^\infty \exp \left[ \frac{1}{n}\frac{2q^n }{1-q^n } \right] = \sum_{n=0}^\infty \dots $$

Answer 1

More generally, $$\sum_{n=1}^\infty\frac{a^n}{n}\frac{1}{1-q^n}=\sum_{l=0}^\infty\sum_{n=1}^\infty\frac{a^nq^{nl}}{n}=\sum_{l=0}^\infty\log\frac{1}{1-aq^l}=\log\frac{1}{(a,q)_\infty}.$$

In particular this says your expression is $(q,q)_\infty^{-2}$. The symbol $(q,q)_\infty^{-1}$ is the generating function for the partition function $p(n)$, and we can interpret its square in a variety of ways (self-convolution of partition function for instance) listed in deoxygerbe's OEIS link or in notes on generating functions, so-called species, or related theory in analytic combinatorics.

Answer 2

Using mpmath:

>>> def g(q):
...     return nprod(lambda n: exp(2*(q**n)/(n*(1-q**n))), [1,inf])
... 


>>> taylor(lambda q: g(q), 0, 20)
[mpf('1.0'), mpf('2.0'), mpf('5.0'), mpf('10.0'), mpf('20.0'), mpf('36.0'),        
mpf('65.0'), mpf('110.0'), mpf('185.0'), mpf('300.0'), mpf('481.0'),     
mpf('752.0'), mpf('1165.0'), mpf('1770.0'), mpf('2665.0'), mpf('3956.0'),   
mpf('5822.0'), mpf('8470.0'), mpf('12230.0'), mpf('17490.0'), mpf('24842.0')]

The coefficients are oeis:A000712 Number of partitions of n into parts of 2 kinds. . In q-Pochhammer symbols, $1/(q;q)^{2}_{\infty}$