Let $(\mathcal A,\varphi)$ be a unital noncommutative probability space with unit $1_{\mathcal A}$ and $\varphi(1_{\mathcal A})=1$, and for each $m\in\mathbb N$ let $\kappa_m:\mathcal A^m\to\mathbb C$ denote the $m$-th free cumulant functional associated to $\varphi$. Let $s=s^*\in\mathcal A$. Then $s$ is a standard semicircular element, meaning $s=s^*$ and

\begin{align*} \varphi(s^m)=0 \quad \text{for odd } m\in\mathbb N \end{align*}

and

\begin{align*} \varphi(s^{2r})=C_r \quad \text{for every } r\in\mathbb N\cup\{0\}, \end{align*}

where $s^0=1_{\mathcal A}$, $C_0=1$, and for $r\geq 1$ the number $C_r$ is the number of noncrossing pairings of $\{1,\dots,2r\}$, if and only if

\begin{align*} \kappa_1(s)=0,\qquad \kappa_2(s,s)=1, \end{align*}

and

\begin{align*} \kappa_m(s,\dots,s)=0 \quad \text{for every } m\in\mathbb N \text{ with } m\ne 2. \end{align*}