1\le a,b\le N-1.\end{gathered}$$

Let us compute the structure constants $(c^{\mathrm{ext}}_v)^{N+1}_{a,b}$. Introduce polynomials $\omega_k\in\mbQ[v_1,\ldots,v_{N-1}]$, $k \ge 0$, by $$\begin{gathered}
\omega_k := \sum_{\substack{\alpha_1,\dots,\alpha_{N-1}\ge 0\\ \sum (N-i)\alpha_i = k}} s_1^{\alpha_1}\cdots s_{N-1}^{\alpha_{N-1}} \frac{(\sum \alpha_i )!}{\prod \alpha_i!},\quad\text{where $s_i:=(1-i)v_i$ for $1\le i\le N-1$.}\end{gathered}$$ The first few functions $\omega_k$ are $$\omega_0 = 1, \quad \omega_1 = s_{N-1}, \quad \omega_2 = s_{N-2} + s_{N-1}^2, \quad \omega_3 = s_{N-3} + 2 s_{N-2}s_{N-1} + s_{N-1}^3.$$ The functions $\omega_k$ satisfy the recursion relation $$\begin{gathered}
\label{eq:recursion for omegak}
\omega_{k+1}=\sum_{i=1}^{N-1}s_{N-i}\omega_{k+1-i},\quad k\ge 0,\end{gathered}$$ where we adopt the convention $\omega_j:=0$ for $j<0$.

\[lemma: Dn xp product\] For $1\le a,b\le N-1$ we have $$\begin{gathered}
\label{eq:cabN+1 in Dn}
c_{a,b}^{N+1} = \sum_{k=0}^{a+b-N-1} \omega_k \frac{v_{N+1}^{2(a+b)-2N-1-2k}}{2^{a+b-N-k}}.\end{gathered}$$

Equation  is equivalent to the following formula: $$\label{eq:Dn cab classes}
{\mathrm{Coef}}_{[w]}[x^{p-2}] = \sum_{k=0}^{p-N-1} \omega_k \frac{v_{N+1}^{2p-2N-1-2k}}{2^{p-N-k}},\quad 2\le p\le 2N-2,$$ where ${\mathrm{Coef}}_{[w]}[x^{p-2}]$ denotes the coefficient of $[w]$ in the expression for $[x^{p-2}]$ in terms of the basis elements $[1],[x],\ldots,[x^{N-2}],[y],[w]$. For $p\le N$ formula  is obvious because both sides of it are equal to zero.

Suppose $p\ge N+1$. Multiplying both sides of equation  by $[x^{p-N-1}]$, we get the relation $${\mathrm{Coef}}_{[w]}[x^{p-2}]=\frac{v_{N+1}^{2p-2N-1}}{2^{p-N}}+\sum_{k=1}^{N-1}s_k{\mathrm{Coef}}_{[w]}[x^{p+k-N-2}],$$ that allows to compute the coefficients ${\mathrm{Coef}}_{[w]}[x^{p-2}]$ recursively. Then, using also relation , formula  can be easily proved by induction.

Using the lemma, we see that equation  