This is symmetric across each axis. We compute that $|\mathcal{S}|=3^d-1$, and $s^{(j)}=3^{d-1}$ for all $j$ and so $$s_n \sim \left( \frac{(3^d-1)^{d/2}}{3^{d(d-1)/2}\cdot\pi^{d/2}}\right) \cdot n^{-d/2} \cdot (3^d-1)^n.$$

Let $e_k = (0,\dots,0,1,0,\dots,0)$ be the $k^\text{th}$ standard basis vector in $\mathbb{R}^d$, and consider the set of steps . Then the number of walks of length $n$ taking steps from ${\ensuremath{\mathcal{S}}}$ and never leaving the positive orthant has asymptotic expansion $$s_n \sim \left(\frac{2d}{\pi}\right)^{d/2}\,n^{-d/2} \, (2d)^n.$$

The second main result is a comparable statement for excursions, Theorem \[thm:excursion\].

Let ${\ensuremath{\mathcal{S}}}\subseteq \{-1,0,1\}^d \setminus \{\mathbf{0}\}$ be a set of unit steps in dimension $d$. If ${\ensuremath{\mathcal{S}}}$ is symmetric with respect to each axis, and ${\ensuremath{\mathcal{S}}}$ takes a positive step in each direction, then the number of walks $e_n$ of length $n$ taking steps in ${\ensuremath{\mathcal{S}}}$, beginning and ending at the origin, and never leaving the positive orthant satisfies $$e_n = O\left(\frac{|{\ensuremath{\mathcal{S}}}|^n}{n^{3d/2}}\right).$$

Organization of the paper
-------------------------

The article is organized as follows. Section \[sec:lwalks\] describes how to express the generating function using an orbit sum by applying the kernel method, following the strategy described in [@BoMi10]. We then derive Equation , which describes the generating function as the diagonal of a rational power series in multiple variables. Section 