A conjecture by the late Romanian mathematician Alexandru Lupas.
Posted in [sci.math][1] in 2005, but no proof was found.
Physicist [Alan Sokal][2] just reminded me of it, saying it was related to something he is working on.

Let $P_n(z)$ be the Legendre polynomials, defined by the generating function
$$
\big(1-2tz+t^2\big)^{-1/2} = \sum_{k=0}^\infty t^k P_k(z) .
$$
Let $g(\alpha,\beta) = 4\cos(2\alpha)+3\sin(\beta)\cos(\alpha)+5$ be defined for 
$(\alpha,\beta) \in (-\pi,\pi)\times (-\pi,\pi)$ .  Let $A_n$ be these "Apéry constants":
$$
A_n = \sum_{k=0}^n \binom{n}{k}\binom{n+k}{k}\sum_{j=0}^k \binom{k}{j}^3 .
$$
  
**examples**  
$$
P_0(z)=1;\qquad
P_1(z)=z;\qquad
P_2(z)=\frac{3}{2}\;z^2-\frac{1}{2};\qquad
P_3(z)=\frac{5}{2}\;z^3-\frac{3}{2}\;z;
\\
A_0 = 1;\qquad A_1=5;\qquad A_2=72;\qquad A_3=1445;\qquad A_4=33001 .
$$

**Prove or disprove:**  
$A_n$ is the average of $P_n\circ g$.  More explicitly:  for all natural numbers $n$,
$$
A_n = \frac{1}{4\pi^2}\int_{-\pi}^\pi \int_{\pi}^\pi 
P_n\big(g(\alpha,\beta)\big)\;d\beta\;d\alpha
$$
  
This is surely true (Sokal says he has checked it through $n=123$).  But can **you** prove it?


  [1]: http://mathforum.org/kb/message.jspa?messageID=3704516
  [2]: http://www.physics.nyu.edu/faculty/sokal/