${\left({}^{2n}C_0\right)}^2-{\left({}^{2n}C_1\right)}^2+{\left({}^{2n}C_2\right)}^2-···+{\left({}^{2n}C_{2n}\right)}^2$

$\mathrm{S}={\left({}^{2\mathrm{n}}\mathrm{C}_0\right)}^2-{\left({}^{2\mathrm{n}}\mathrm{C}_1\right)}^2+{\left({}^{2\mathrm{n}}\mathrm{C}_2\right)}^2-···+{\left({}^{2\mathrm{n}}\mathrm{C}_{2\mathrm{n}}\right)}^2$

we know that ${{\mathrm{C}}_0}^2-{{\mathrm{C}}_1}^2+{{\mathrm{C}}_2}^2+···{(-1)}^2 {{\mathrm{C}}_{\mathrm{n}}}^2={(-1)}^{\frac{n}{2}} n_{C_{\frac{n}{2}}}$  if n is even.

$\therefore \mathrm{S}={(-1)}^{\mathrm{n}} {}^{2\mathrm{n}}{\mathrm{C}}_{\mathrm{n}}$