Let $H$ be a complex [Hilbert space](/page/Hilbert%20Space), let $\mathcal{F}(H)$ be the full Fock space with vacuum vector $\Omega$, and let $\tau$ be the vacuum expectation state defined by $\tau(T) = (T\Omega,\Omega)_{\mathcal{F}(H)}$ for bounded operators $T \in \mathcal{L}(\mathcal{F}(H))$. For $f \in H$ with $\|f\|_H = 1$, let $\ell(f) \in \mathcal{L}(\mathcal{F}(H))$ denote the left creation operator and let

\begin{align*} s(f) := \ell(f) + \ell(f)^*. \end{align*}

Then, for every integer $m \geq 0$,

\begin{align*} \tau(s(f)^{2m+1}) = 0 \end{align*}

and

\begin{align*} \tau(s(f)^{2m}) = C_m, \end{align*}

where the $m$-th Catalan number is

\begin{align*} C_m = \frac{1}{m+1}\binom{2m}{m}. \end{align*}