Generating function for $\sum{\sqrt{n^2+n}x^na_n}$ if $\sum x^{n+1}a_{n+1} = A(x)$

Question

I am looking for a a simplification for $\sum{\sqrt{n^2+n}x^na_n}$ if $\sum x^{n+1}a_{n+1} = A(x)$ (summing over $n$). A problem that uses something similar to this is in Generatingfunctionology and I am trying to get a more general solution.