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This is not an answer but a conjecture. It based on a first examples examples calculated by Mathematica. $$S(a)=\frac1{a+1}-\sum_{i=1}^a\frac{\psi(-\alpha_i)}{f_a'(\alpha_i)},$$ where $f_0(x)=1$, $f_1(x)=x+3$, $f_2(x)=x^2+5x+7$,... $$f_{n+1}(x)=(x+2)f_n(x)+1\qquad(n\ge 0),$$ (see A193844) and $\alpha_i$ are the roots of $f_a$.