Okay I'm not sure if this is University level or below, but anyway

Apply the Leibnitz theorem to the function $\displaystyle y=x^{2n}$

and hence prove that,

$\displaystyle 1 + (n^2)/1 + ((n^2)(n-1)^2)/(1^2)(2^2) + ........ = (2n!)/(n!)^2$

Now I can easily prove this using the Binomial theorem as the terms on the LHS are merely squares of the binomial coefficients in the expansion $\displaystyle (1+x)^n$

And the RHS is the coefficient of $\displaystyle x^n$ in the expansion of

$\displaystyle (1+x)^{2n}$

Using this data the above relation can be proved easily.

But I am not able to prove it using the method the question demands, that is, by using the Lebnitz formula for derivative of a function.

Can anyone help?