Let $\Psi$ be the maps \begin{array}{lrcl} \Psi : & {\mathbb N}^{(\mathbb N)} & \longrightarrow & \mathbb R^+\cup\{+\infty\} \\ & A=(a_i)_{i\in \{0,\ldots,n-1\}} & \longmapsto & \displaystyle\int_{\mathbb R^+} \displaystyle\frac{16x^2}{\sqrt{\displaystyle\sum_{i=0}^n a_ix^{2i}}}\text d x. \end{array}
We denote by $\mathfrak P$ the subset of ${\mathbb N}^{(\mathbb N)}$ so that $$\forall A\in \mathfrak P \quad \Psi(A)=\pi.$$
We know that $\mathfrak P\ne \emptyset$, because $\Psi((1,6,35,15,6,1))=\pi$.
What else can be said about $\mathfrak P$ ?
Is there a way to characterize all elements of $\mathfrak P$ ?
Is there any interesting necessary or sufficient conditions to be in $\mathfrak P$ ?
Does $\mathfrak P$ admits elements of any lengths (big enough) ?
Is $\mathfrak P$ even infinite ?
Is there any palindromes in $\mathfrak P$ ?