Post History

Do the Faber partition polynomials have integer coefficients?

The Online Encyclopedia of Integer Sequences includes [A263916: Coefficients of the Faber partition polynomials](https://oeis.org/A263916). Perhaps the clearest definition given is

> -log(1 + b(1) x + b(2) x^2 + ...) = Sum_{n>=1} F(n,b(1),...,b(n)) * x^n/n

which in better notation is

$$\sum_{n \ge 1} F_n(b_1, \ldots, b_n) \frac{x^n}n = -\log(1 + b_1 x + b_2 x^2 + \cdots)$$

E.g. $$F_4(b_1,b_2,b_3,b_4) = -4b_4 + 4b_1 b_3 + 2b_2^2 - 4b_1^2 b_2 + b_1^4$$

The fact that it's listed in OEIS suggests that all of the coefficients should be integers, but none of the formulae or comments in the page obviously tells me that they are. What's the most straightforward way to see this?