After a series of calculations following this question of mine (Constant term of noncommutative $(X+Y+(XY)^{-1})^n$), I've cooked up the following function: $$F_n(q)=(n+1)^{-n}\frac{(q+(n+1))^n}{q+1}$$ Where $q$ is the formal power series in $u$ satisfying the functional equation: $$u=q\frac{(q+(n+1))^n}{(n+1)^{(n+1)}(q+1)^{(n+1)}}.$$ The claim is that the above function is an (algebraic, this is clear) Hypegeometric function. I've made a great amount of tries on how to prove this, but none of them has led anywhere. The only thing I have managed to write, is a mathematica program which allows me to calculates any number of first terms, a few examples are: \begin{align*} & 1 -t^3 -3t^6 -17t^9 -123t^{12} -1017t^{15} -9155t^{18} -87441t^{21} -872523t^{24} + \dots,\quad \text{for } n=2 \\ & 1 -t^4 -6t^8 -68t^{12} -997t^{16} -16860t^{12} -312392t^{16} -6169036t^{20} + \dots,\quad \text{for } n=3 \\ & 1 -t^5 -10t^{10} -190t^{15} -4705t^{20} -134999t^{25} -4256420t^{30} + \dots,\quad \text{for } n=4 \\ & 1 -t^{6} -15t^{12} -430t^{18} -16140t^{24} -703794t^{30} -33777171t^{36} + \dots,\quad \text{for } n=5 \\ & 1 - t^7 - 21 t^{14} - 847 t^{21} - 44870 t^{28} - 2766078 t^{35} - 187861254 t^{42} + \dots,\quad \text{for } n=6 \end{align*} (if needed I can provide the code)
Since the claim that this is a Hypergeometric function is equivalent to finding a rational function of $n$ which interpolates with the quotients at each positive integer of the successive coefficients. And since these quotients seem to stabilize after some time, it seems like a plausible result.
Unortunately I have no idea on how to compute a rational interpolation, maybe someone who knows how to deal with such problems could help me out...
Many thanks in advance. Any help is greatly appreciated.