Given an arbitrary partition $\lambda$ and an integer $N$ (the number of variables), is there any further way to evaluate the following derivative of the Schur polynomial?

\begin{align} A &= \frac{1}{2} \left.\left(x \frac{d}{dx}\right)^2 s_\lambda(x, x^{-1}, \underbrace{1, \dotsc, 1}_{N-2})\right|_{x = 1} \\ &= \frac{1}{N (N - 1)} \sum_{\sigma \in S_{\lambda_1}} \mathop{\mathrm{sgn}}(\sigma) \sum_{j = 1}^{\lambda_1} (\lambda'_j - j + \sigma(j)) (N - \lambda'_j + j - \sigma(j)) \prod_{i = 1}^{\lambda_1} \binom{N}{\lambda'_i - i + \sigma(i)} \end{align}

The expression given is found just from explicitly differentiating the dual Jacobi–Trudi expression for the Schur polynomial.