Diferenciando $\left[\frac{\mathrm{d}^{n}}{\mathrm{d}x^{n}}(x^{2}-1)^{n}\right]^{2}$

Question

Dado que $$\frac{\mathrm{d}^{2n}}{\mathrm{d}x^{2n}}(x^{2}-1)^{n} = (2n)!$$

¿Cómo podemos encontrar $$\left[\frac{\mathrm{d}^{n}}{\mathrm{d}x^{n}}(x^{2}-1)^{n}\right]^{2}\quad ?$$

Answer 1

$\left(\dfrac{d^n}{dx^n}(x^2-1)^n\right)^2$

$=\biggl(\dfrac{d^n}{dx^n}\sum\limits_{k=0}^nC_k^n(-1)^kx^{2n-2k}\biggr)^2$

$=\biggl(\sum\limits_{k=0}^n\dfrac{(-1)^kn!(2n-2k)!x^{n-2k}}{k!(n-k)!(n-2k)!}\biggr)^2$

$=\biggl(\sum\limits_{k=0}^\left[\frac{n}{2}\right]\dfrac{(-1)^kn!(2n-2k)!x^{n-2k}}{k!(n-k)!(n-2k)!}\biggr)^2$