A short proof of Mathar's 2013 recurrence conjecture for the reversible-binary-string sequence A032123

For the OEIS sequence A032123, the number of length-$2n$ black-and-white strings with $n$ black beads, considered up to reversal, R. J. Mathar contributed in November 2013 the conjectured order-5 P-recursive recurrence $$\begin{aligned} &n(n-1)\,a(n) - 2(n-1)(3n-4)\,a(n-1) + 4(2n^{2}-14n+19)\,a(n-2) &\qquad + 8(n^{2}+5n-19)\,a(n-3) - 16(n-3)(3n-10)\,a(n-4) &\qquad + 32(n-4)(2n-9)\,a(n-5) \;=\; 0, \qquad n \ge 6. \end{aligned}$$ We give a short proof. Burnside's lemma applied to the reversal action gives the closed form $a(n) = \tfrac{1}{2}\bigl(\binom{2n}{n} + [n \text{ even}]\binom{n}{n/2}\bigr)$; the two summands satisfy elementary recurrences of order $1$ and $2$ respectively; and Mathar's order-5 operator, applied to each summand separately, reduces to a polynomial identity that simplifies to zero after a brief calculation. The supplementary archive includes a SymPy script which verifies the polynomial identities symbolically and checks Mathar's recurrence numerically for $n = 6, \ldots, 5000$.