While writing a program to compute bases of spherical harmonics on $S^n$, I discovered that the following array of rational numbers naturally arises when one considers the inclusion $S^n \subset S^{n+1}$:
\begin{array}{ccccccc} \frac{2}{3} & \frac{8}{15} & \frac{16}{35} & \frac{128}{315} & \frac{256}{693} & \frac{1024}{3003} & \cdots \\ \frac{3}{4} & \frac{5}{8} & \frac{35}{64} & \frac{63}{128} & \frac{231}{512} & \frac{429}{1024} & \cdots \\ \frac{4}{5} & \frac{24}{35} & \frac{64}{105} & \frac{128}{231} & \frac{512}{1001} & \frac{1024}{2145} & \cdots \\ \frac{5}{6} & \frac{35}{48} & \frac{21}{32} & \frac{77}{128} & \frac{143}{256} & \frac{2145}{4096} & \cdots \\ \frac{6}{7} & \frac{16}{21} & \frac{160}{231} & \frac{640}{1001} & \frac{256}{429} & \frac{4096}{7293} & \cdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{array}
Each entry $a_{ij}$ of this array is computed by taking a spherical harmonic on $S^i$ of degree $j$ and computing its $L^2$ norms with respect to the normalized surface area measures on $S^{i}$ and $S^{i+1}$. The ratio of these two norms turns out to be independent of the spherical harmonic one starts with, and $a_{ij}$ is the square of this ratio.
There are some clear and striking patterns here, but I haven't been able to guess a formula for $a_{ij}$ or relate this array to any well-known integer sequences on OEIS. Any ideas?
By sheer luck, I managed to stumble upon the correct formula shortly after posting this question.
$$ a_{ij} = \frac{i!!}{(i-1)!!} \cdot \frac{(i+2j-1)!!}{(i+2j)!!} $$
Here, $n!!$ denotes the double factorial.