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

Abstract

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