A short proof of Mathar's 2013 recurrence conjecture for the Laguerre sequence~A025166

For the OEIS sequence A025166, defined by $a(n)=-n!2^n{L}_n(1\mathrm{/}2)$ where $L_n$ is the Laguerre polynomial of degree $n$, R.~J.~Mathar contributed in February 2013 the conjectured order-2 P-recursive recurrence $$a(n)+(-4n+3)a(n-1)+4(n-1{)}^{2}a(n-2)=0,n\ge 2.$$ We give a one-page proof. The exponential generating function $F(x)=-\exp \text{\hspace{0.17em}⁣}(-x\mathrm{/}(1-2x))\mathrm{/}(1-2x)$ satisfies the first-order linear ODE $(1-2x{)}^2{F}^{\mathrm{\prime }}(x)=(1-4x)F(x)$, and Mathar’s recurrence then falls out by reading off the coefficient of $x^n\mathrm{/}n!$. Both steps are short. The supplementary archive includes a SymPy script which checks the ODE identically and the recurrence numerically up to $n=5000$.