A short proof of Mathar's 2013 recurrence conjecture for the Meixner sequence A214615

For the OEIS sequence A214615, defined by $a(n)=M_n(1)$ where $M_n$ is the $n$-th Meixner polynomial satisfying $M_{n+1}(x)=x{M}_n(x)-n^2{M}_{n-1}(x)$, R.~J.~Mathar contributed on 6~March 2013 the conjectured order-2 P-recursive recurrence $a(n)-a(n-1)+(n-1{)}^{2}a(n-2)=0$ for $n\ge 2$. We give a one-page proof. The exponential generating function $F(t)=\exp \text{\hspace{0.17em}⁣}(\arctan t)\mathrm{/}\sqrt{1+t^2}$ satisfies the first-order linear ODE $(1+t^2)F^{\mathrm{\prime }}(t)=(1-t)F(t)$, and Mathar’s recurrence then falls out by reading off the coefficient of $t^n\mathrm{/}n!$. Both steps are short. The supplementary archive includes a SymPy script that checks the ODE identically and the recurrence numerically up to $n=500$.