Positivity of the coefficients of the Ehrhart polynomial of a cross-polytope

Here is a very simple way to show the positivity. Define $$f(d,x) = (1+x)^d/(1-x)^{d+1}.$$ Then, by induction $$ \frac{\partial^t f(d,x)}{\partial\, d^t} = f(d,x) \, \ln\biggl(\frac{1+x}{1-x}\biggr)^t.$$ Putting in $d=0$ we have that the Taylor series of $f(d,x)$ with respect to $d$ is $$ f(d,x) = \sum_{t=0}^\infty \frac{1}{t!} (1-x)^{-1} \ln\biggl(\frac{1+x}{1-x}\biggr)^t\,d^t. $$ Both $(1-x)^{-1}$ and $\ln\Bigl(\frac{1+x}{1-x}\Bigr)$ have non-negative Taylor coefficients, which completes the proof.

In summary, the coefficient of $x^nd^t$ is $2^t/t!$ times the coefficient of $x^n$ in $$\biggl(\sum_{k\ge 0} x^k\biggr) \biggl(\sum_{k\ge 0} \frac{x^{2k+1}}{2k+1}\biggr)^t. $$