Let us consider polynomials as functions on $[0,1]$, and so define 
\begin{align*}
 \|f\|_2 &= \sqrt{\int_0^1f(x)^2\,dx} \\
 \|f\|_\infty &= \max\{|f(x)|: 0 \leq x\leq 1\}.
\end{align*}
Put 
$$ f_d(x) = \sum_{i=0}^d \frac{(d+1+i)!}{(d-i)!i!(i+1)!}(-x)^i. $$
Experiments make it clear that $\|f_d\|_2=1$ and $\|f_d\|_\infty=(d+1)$ and that $f_d$ maximises the ratio $\|f\|_\infty/\|f\|_2$ among polynomials of degree $d$.  These facts must surely be known.  Can anyone point me to a reference?  Do the polynomials $f_d(x)$ have a standard name?