Legendre polynomial with argument zero $P_n ( 0 )$ I want to find the expression for $P_n(x)$ with $x = 0$, ie $P_n(0)$ for any $n$. The first few non-zero legendre polynomials with $x=0$ are 
$P_0(0) = 1$, $P_2(0) = -\frac{1}{2}$, $P_4(0) = \frac{3}{8}$, $P_6(0) = -\frac{5}{16}$, $P_8(0) = \frac{35}{128}$ but I can't find a relationship between them to write as an equation for arbitrary $n$. Any help is appreciated.


See Proving a property of Legendre Polynomials where the following formula is proven:
\begin{equation}
\begin{split}
P_{2n}(0) &=\frac{1}{2^{2n}}\sum_{k=0}^{2n}\binom{2n}{k}^2(-1)^k = \frac{1}{2^{2n}}\binom{2n}{n}(-1)^n = \frac{(-1)^n(2n)!}{2^{2n}n!^2}
\end{split}
\end{equation}
