We denote by $n$ the integer part of $\frac{N}{2}$. For $j \in \mathbb{N}$, the polynomials $P_j$ are defined by $P_{j}(X) = \frac{1}{2^{j} j!} \frac{d^{j}}{dX^{j}}\left[(X^{2}-1)^{j}\right]$, and $g_j = \int_{-1}^{1} P_j(x)^2\,dx$.
Discuss, depending on the parity of $n$, the value of $a_{N}$. We will give its explicit value.