Binomial expansion comparison with legendre polynomial

Hi,

I've been working on this question which asks to show that

So first taking the n derivatives of the binomial expansions of (x^{2}-1)^{n }

and comparing it with

I'm having trouble with the final part,

It's clear that there's a factor of 1/n!2^{n }difference between them but also

the Pn(x) series has m=0...n/2, and also x^{n} , where as the n'th derivative series has k=0...n and x^{2n}.

How can you rewrite one in terms of the other so they both have the same sum limits?

I've tried setting k=2s in the n'th derivative series and a bunch of other similar changes, but non will change the n'th powers of x.

The reason I noticed this was because the last terms of the series arn't the same,

the first series has last term, (-n)! on the bottom, means 1/infinity right?

and the second

Have I made a mistake early on or is there a clever way to combine the two series?

Thanks,

Linda