Sums of squares and sequences of modular forms

Let hₙ(v) be the sequence of rational functions with hₙ(v)/v-nhₙ(v)+(n-1)hₙ₋₁(v)-vhₙ₋₁'(v)+v(v(vhₙ₋₁(v))')'4=0 for $n>0$ and h₀(v)=1. We prove that hₙ(v) has a pole at v=1/n if and only if n is a sum of two squares of integers. Moreover, if r₂(n)=#\(a,b)∈ Z²: a²+b²=n\, then we derive the formula v=1/nReshₙ(v)=(-1)ⁿ⁻¹r₂(n)n16ⁿ. The results are then generalized to arbitrary modular forms with respect to Γ(2) and as a consequence we obtain a new criterion for Lehmer's conjecture for Ramanujan's τ-function.