Orthorecursive expansion of unity

Define the sequence $c_n$ by relations
$$ c_0=1, \quad \frac{c_0}{n+1}\,+\,\ldots\,+\,\frac{c_n}{2n+1}\,=\,0 $$
for all $n>0$. Despite simple definition, this sequence has interesting properties and turns out to be connected with orthorecursive expansions in the space $L^{2}[0,1]$. In my talk, I will discuss these properties (some of them are proved and some are observed experimentally) and tell you how permutations of the set of $n$ elements help us to prove that $c_n\neq 0$.