Thank you @Carlo. I posted an example: physics.stackexchange.com/questions/358631/…

@AmirSagiv This is equation for ground state of the Gutzwiller bosonic wavefunction when you deal with hopping hamiltonian - cold atoms in optical lattice.

Sorry, I missed extra index in $\alpha$. Coefficients $\alpha_{i,n}$ depend on both $n$ and $i$.

@RobertBryant Indeed there was a mistake in recursion relation. This is a physics problem and as you noticed coefficient $c_{n,m}$ is linked with others such that $s=n+m$ is fixed. We can of course find eigenvalues for small $s$ and find time evolution explicitly. However this doesn't solve my problem cause imagine I want to know what happens when $s$ is large and theoretically tends to infinity.