In a calculation of some momenta of random matrices (GOE), I encounter a functional equation, in the form of a second-order recursion,
$$L(s+1)=L(s)+2s(2s+1)L(s-1).$$
Is it familiar to someone ? Is there an explicit formula for the solutions ? By explicit, I mean in terms of the Gamma function for instance.

Just in case, I am interested in the solution generated by $L(0)=1$ and $L(1)=-\frac18$ .