Using the recursion formula for the solution of the Legendre equation:
$$(1-x^2)y''(x)-2xy'(x)+n(n+1)y(x) = 0$$
With solution $P_n(x)$ such that $P_n(1) = 1$, show that
$$P_n(x) = \sum_{k = 0}^{n}\frac{(-1)^k}{2^n}{n\choose k}{2n-2k\choose n}x^{n-2k}$$
So I have the recursive formula for the next element in the polynomial:
$$a_{k+2} = \frac{k(k+1)-n(n+1)}{(k+1)(k+2)}a_k$$
And the formulas for each coefficient based on $a_0,a_1$:
$$a_{2m} = (-1)^m\frac{\prod_{k = 0}^{2m-1}(n+(-1)^{k+1}k)}{k!}a_0$$ $$a_{2m+1} = (-1)^m\frac{\prod_{k = 1}^{2m}(n+(-1)^{k}k)}{k!}a_1$$
But I don't know how to procede from here. I don't even know how to define $a_0$ and $a_1$