Let $f_i=f_i(x_1,x_2,\ldots, x_n),i=0,1,2, \ldots $ be a family of symmetric polynomials. For the partition $\lambda=(\lambda_1,\lambda_2, \ldots, \lambda_n)$ consider the determinant
$$
D_\lambda(f)=\left | \begin{array}{lllll}
f_{\lambda_1} & f_{\lambda_1+1} & f_{\lambda_1+2} & \ldots & f_{\lambda_1+n-1}\\
f_{\lambda_2-1} & f_{\lambda_2} & f_{\lambda_2+1} & \ldots & f_{\lambda_2+n-2}\\
\vdots & \vdots & \vdots & \ldots & \vdots \\
f_{\lambda_n-(n-1)} & f_{\lambda_n-(n-2)} & f_{\lambda_n-(n-3)} & \ldots & f_{\lambda_n}
\end{array} \right|.
$$
It is well known (Jacobi−Trudi formulas) that for the elementary symmetric polynomials $e_i=e_i(x_1,x_2,\ldots, x_n)$ and for the complete homogeneous symmetric polynomials $h_i=h_i(x_1,x_2,\ldots, x_n)$ we have 
$$
D_\lambda(h)=s_{\lambda}(x_1,x_2,\ldots, x_n) \text{ and } D_\lambda(e)=s_{\lambda'}(x_1,x_2,\ldots, x_n), 
$$
where $s_{\lambda}(x_1,x_2,\ldots, x_n)$ is the Schur polynomial and $\lambda'$ is the conjugate partition.

**Question.** Is there a similar expression for $D_\lambda(p)$ where $p_i=p_i(x_1,x_2,\ldots, x_n)$ is the power sum symmetric polynomials?

By direct calculation for $n=2, \lambda_2\geq 1$ I got that 

$$D_{(\lambda_1,\lambda_2)}(p)=-s_{(\lambda_1-1,\lambda_2-1)} V(x_1,x_2)^2$$ and for $n=3,\lambda_3\geq 2$ 

$$D_{(\lambda_1,\lambda_2,\lambda_3)}(p)=-\frac{s_{(\lambda_1-1,\lambda_2-1,\lambda_3-1)}}{s_{(1,1,1)}} V(x_1,x_2,x_3)^2$$
and so on.

 Here $V$ is the Vandermonde determinant.

I hope that must be a nice known formula and for any $n$ but can't find it.