Sums of squares and sequences of modular forms

Let $h_n(v)$ be the sequence of rational functions with $$\frac{h_n(v)}{v}-n{h}_n(v)+(n-1)h_{n-1}(v)-v{h}_{n-1}^{\mathrm{\prime }}(v)+\frac{v(v(v{h}_{n-1}(v){)}^{\mathrm{\prime }}{)}^{\mathrm{\prime }}}{4}=0$$ for $n>0$ and $h_0(v)=1$. We prove that $h_n(v)$ has a pole at $v=\frac{1}{n}$ if and only if $n$ is a sum of two squares of integers. Moreover, if $r_2(n)=\mathrm{\#}\{(a,b)\in {\mathbb{Z}}^2:a^2+b^2=n\}$, then we derive the formula $$h_n(v)=\frac{(-1{)}^{n-1}{r}_2(n)}{n{16}^n}\mathrm{.}$$ The results are then generalized to arbitrary modular forms with respect to $\mathrm{\Gamma }(2)$ and as a consequence we obtain a new criterion for Lehmer’s conjecture for Ramanujan’s $\tau$-function.