Asymptotics for an exponential generating function from an ordinary

Question

I'm interested in taking an ordinary generating function $$F(x)=\sum_{n\geq 1}m_nx^n$$ and converting it to an exponential generating function $$M(x)=\sum_{n\geq 1}m_n\frac{x^n}{n!}.$$ I would then like to say something about the asymptotics of the sequence of numbers $c_n$ defined by $$\sum_{n\geq 1}c_n\frac{x^n}{n!}=\log(1+M(x)).$$ My question is if there is any method for doing this. The specific example I have in mind is the function $$F(x)=\frac{-1-x-\sqrt{1-14x+x^2}}{8}.$$

I know that one can convert from ordinary generating functions to exponential generating functions using the inverse Laplace transform $\mathcal L^{-1}$. Namely, I should have $$M(x)=\mathcal L^{-1}\{F(1/z)/z\}(x).$$ Unfortunately, I don't think I can get a closed-form expression for this inverse Laplace transform in the case I'm considering.

Answer 1

Using the gfun package in Maple, it appears that

$$ M(x) = -\frac{1}{4}-{ \frac {3\,x}{28}}-6 (x-7)\,\int_{0}^{x}\!{\frac {{{\rm e}^{t \left( 4\,\sqrt {3}+7 \right) }}}{ \left( t-7 \right) ^{2}}{\it HeunC} \left( 56\,\sqrt {3} ,1,-2,0,-{\frac{95}{2}},\frac{t}{7} \right) }\,{\rm d}t $$