I'm studying Frobenius Manifolds associated with $A_n$-type singularities and in order to prove some results about their potentials I need to calculate the following thing.
Assume that $n$ is a fixed natural number and there is a finite family of $n-1$ polynomials with rational coefficients on $n-1$ variables $f_q(x_1,\dots,x_{n-1})$ for $1\leqslant q \leqslant n-1$, whose coefficients and the polynomials themselves are defined recursively. Moreover, each polynomial is defined as a sum over all tuples of natural numbers which subject to a condition on their sum. The main question is the following: how to calculate partial derivatives (actually I need only second ones, i.e. a Hessian matrix) of these objects?
Maybe some more explicit details could help to calculate derivatives, so let me define the family which I am interested in.
Firstly, let me define recursively the numbers $P_n(s_1,\dots,s_m)$ for a fixed natural number $n$ and an $m$-tuple of natural numbers $(s_1,\dots,s_m)$ as follows: $$P_n(s_1)=s_1$$ $$P_n(s_1,\dots,s_m) = \binom{n}{m} - \sum\limits_{q=1}^{m-1} P_n(s_1,\dots,s_q) \binom{n-q-(s_1+\cdots+s_q)}{m-q}$$
where I assume that $\binom{p}{t} = 0$ are defined as usual for $p\geqslant t \geqslant 0$ and $0$ otherwise. For instance, $P_n(s_1,\dots,s_m) = 0$ for $m\geqslant n+1$.
Now I define the main object, namely a family of polynomials with rational coefficients $x_{-q}(x_1,\dots,x_{n-1})$ ($n$ is still a fixed natural number) for $q \geqslant 0$ as follows:
$x_{0} = 0$ and $x_{-q}(x_1,\dots,x_{n-1})$ for $q > 0$ is defined recursively via
$$x_{-q} = -\frac{1}{n} \sum\limits_{m=2}^{n}\sum P_n(s_1,\dots,s_m) x_{n-s_1}\cdots x_{n-s_m}$$
where the second sum is taken over all natural numbers $(s_1,\dots,s_m)$ such that $\sum\limits_{i=1}^m s_i = q+n+1-m$.
For example, here is the list of them for $n=3$ (one can also prove that $x_{-n}$ are always equal to zero): $$x_{-1} = -x_1x_2$$ $$x_{-2} = -x_1^2-\frac{x_2^3}{3}$$
Then I need to compute all second partial derivatives of $x_{-q}$ (i.e. Hessian matrix $H_{n,q}$ of $x_{-q}$) for any $n$ and $1\leqslant q \leqslant n-1$ and it is the point where I've been completely stuck. I need these derivatives in order to check some equalities involving them and I am able to found them via direct calculations for some small numbers $n$, but I can't cope with the general case because the coefficients $P_n$ and the polynomials $x_{-q}$ are defined recursively and moreover $x_{-q}$ is defined via the implicit summation condition, that's why I do not understand how to write down a closed formula for any $n$.
Actually I can prove it, but only for $q=1$. In this case it follows from some general theory of Frobenius Manifolds that $-H_{n,1}$ is equal to the pairing matrix for $A_{n-1}$-type Frobenius Manifold which is just an antidiagonal matrix
I'd appreciate any help and thoughts on the computation or even on some properties of the coefficients $P_n$, polynomials $x_{-q}$ or their Hessian matrix.