linear differential equation $2x^2 I'(x) + xI(x)+1 = I(x)$ from which the differential equation $C'(x) = \frac{C(x)(1+C(x))-x}{2x C(x)}$ [@Flajolet2000] can be deduced: $$\begin{aligned}
 \label{eqn:asyC} ({\mathcal{A}}^2_{\frac12} C)(x)&= \frac{1}{\sqrt{2\pi}}\frac{x}{C(x)}e^{-\frac{1}{2 x}(2C(x) + C(x)^2)}. \end{aligned}$$ This is the generating function of the full asymptotic expansion of $C_n$. The first few terms are, $$\begin{aligned}
 ({\mathcal{A}}^2_{\frac12} C)(x) = \frac{e^{-1}}{\sqrt{2 \pi}} \left( 1 - \frac{5}{2} x - \frac{43}{8} x^2 - \frac{579}{16} x^3 - \frac{44477}{128} x^4 - \frac{5326191}{1280} x^5 + \ldots \right). \end{aligned}$$ Expressed in the traditional way using eq.  from Definition \[def:basic\_asymp\_definition\] this becomes $$\begin{aligned}
 \begin{split} C_n &\sim \sum_{k \geq 0} 2^{n+\frac12 -k} \Gamma(n+\frac12 -k) [x^k] ({\mathcal{A}}^2_{\frac12} C)(x) = \sqrt{2 \pi} \sum_{k \geq 0} (2(n-k)-1)!! [x^k] ({\mathcal{A}}^2_{\frac12} C)(x)\\ &= e^{-1} \left( (2n-1)!! - \frac{5}{2} (2n-3)!! - \frac{43}{8} (2n-5)!! - \frac{579}{16} (2n-7)!! + \ldots \right) \end{split} \end{aligned}$$ The first term, $e^{-1}$, in this expansion has been computed by Kleitman [@kleitman1970proportions], Stein and Everett [@stein1978class] and Bender and Richmond [@Bender1984] each using different methods. With the presented method an arbitrary number of coefficients can be computed. Some additional coefficients are given in Table \[tab:coefficientsCM\].

The probability of a random chord diagram with $n$ chords to be connected is therefore $e^{-1}(1- \frac{5}{4n}) + {\mathcal{O}}(\frac{1}{n^2})$.

Monolithic chord diagrams
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A chord 