TITLE: Minimising an expression - involving polynomial
QUESTION [7 upvotes]: I found this one on a forum but it has been unanswered from long there. I am curious to know if there is a solution to this problem. Here it is:
Let n be a positive integer. Determine the smallest possible value of
$$|p(1)|^2+|p(2)|^2 + .........+ |p(n+3)|^2$$
over all monic polynomials p with degree n.

I don't have much idea about the problem. I can go about by assuming $p$ to be some monic polynomial and find $p(1)$, $p(2)$.... in terms of the coefficients but I don't think that is going to help. This is some inequality related problem, I guess.
Any help is appreciated. Thanks!
PS: Can somebody please check if I have added the proper tags? I am not sure where this kind of problem should belong.

REPLY [3 votes]: Cross-posted and answered 
on Mathoverflow. 
The formula is $(2n+1)(2n+3) n!^4 / (2n)!$.
See my answer to 
MO_210823 for the proof.
In general the minimum value of $\sum_{i=1}^{n+k+1} p(i)^2$
over monic polynomials $p$ of degree $n$ is
$$
\frac{n!^4}{(2n)!} {2n+1+k \, \choose k},
$$
and the minimizing $p$ is proportional to a
Hahn polynomial
$Q_n$ evaluated at $x-1$ (with $\alpha=\beta=0$ and $N = n+k+1$).