does not cross any chords. Let $I(x) = \sum_{n=0}(2n-1)!!x^n$ and $C(x) = \sum_{n=0} C_n x^n$, where $C_n$ is the number of connected chord diagrams with $n$ chords. Following [@Flajolet2000], the power series $I(x)$ and $C(x)$ are related by, $$\begin{aligned}
 \label{eqn:cntd_chords} I(x) &= 1+C(xI(x)^2). \end{aligned}$$ This functional equation can be solved for the coefficients of $C(x)$ by basic iterative methods. The first few terms are, $$\begin{aligned}
 C(x) = x + x^2 + 4x^3 + 27x^4 + 248 x^5 +\ldots \end{aligned}$$ This sequence is entry `A000699` in Neil Sloane’s integer sequence on-line encyclopedia [@oeisCntdChords]. Because $(2n-1)!!=\frac{2^{n+\frac12}}{\sqrt{2 \pi}}\Gamma(n+\frac12)$, the power series $I$ is in ${{\mathbb{R}}[[x]]^{2}_{{\frac12}}}$ and $({\mathcal{A}}^2_{\frac12} I)(x) = \frac{1}{\sqrt{2 \pi}}$. Theorem \[thm:chainrule\] guarantees that also $C \in {{\mathbb{R}}[[x]]^{2}_{{\frac12}}}$, because of the closure properties of ${{\mathbb{R}}[[x]]^{2}_{{\frac12}}}$. Moreover, an application of the general chain rule from Theorem \[thm:chainrule\] on the functional eq. results in $$\begin{gathered}
 \begin{gathered} ({\mathcal{A}}^2_{\frac12} I)(x) = {\mathcal{A}}^2_{\frac12} (1+C(xI(x)^2)) = {\mathcal{A}}^2_{\frac12} C(xI(x)^2) \\= 2 x I(x) C'(xI(x)^2) ({\mathcal{A}}^2_{\frac12} I)(x) + \left(\frac{x}{xI(x)^2} \right)^\frac12 e^{\frac{xI(x)^2-x}{2 x^2 I(x)^2}} ({\mathcal{A}}^2_{\frac12} C)(x I(x)^2), \end{gathered} \end{gathered}$$ which can be solved for $({\mathcal{A}}^2_{\frac12} C)(x)$ and simplified using eq. , $$\begin{aligned}
 ({\mathcal{A}}^2_{\frac12} C)(x)&= \frac{1+C(x)-2x C'(x)}{\sqrt{2\pi}}e^{-\frac{1}{2 x}( 2C(x) + C(x)^2 )}. \end{aligned}$$ A further simplification can be achieved by utilizing the 