This is motivated by [this question](https://mathoverflow.net/questions/85719).


Let $f$ be the hypergeometric series


$ f(x) = 2 x \, \_{4}F\_3([1, 1, 4/3, 5/3], [2, 2, 2], 27 x) $


which is explictly given by


$ f(x) = \sum\_{n \geq 1} \frac{(3n-1)!}{n!^3} x^n $


and appears in some studies of Mahler measures.



> 
> Is is true that $\exp(f)$ has only positive integer coefficients ?
> 
> 
> 


The first few coefficients are $1, 2, 17, 218, 3404, 59644$.


If true, is there a combinatorial meaning for these coefficients ?


In the same vein, what can one say on the coefficients of $\exp(-f)$ which start


$1, -2, -13, -158, -2431, -42454, -802790$ ?



Exponential of a specific hypergeometric series
The integrality is a re-incarnation of the problem discussed by Jan Stienstra in ["Mahler Measure Variations, Eisenstein Series and Instanton Expansions"](http://arxiv.org/abs/math/0502193) (particularly eq. (6) there). The point is that the normalised derivative of the function is an explicit modular form after a suitable modular-function parametrisation, and all this integrality and postivity properties can be extracted from this modular structure on demand. I do not see an easy way to produce an explicit answer to the OP, as this really requires some technical work. But if the author (F.C.) is interested, he can certainly reconstruct the technicalities. I would say that this particular integrality/positivity is less challenging than usual, because we can relate the expansions to modular forms.


