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 $$\begin{array}{rlrl}{\displaystyle }& {\displaystyle n(n-1)a(n)-2(n-1)(3n-4)a(n-1)+4(2n^2-14n+19)a(n-2)}& {\displaystyle +8(n^2+5n-19)a(n-3)-16(n-3)(3n-10)a(n-4)}& {\displaystyle +32(n-4)(2n-9)a(n-5)=0,n\ge 6.}\end{array}$$ We give a short proof. Burnside’s lemma applied to the reversal action gives the closed form $a(n)=\frac{1}{2}(\left(\genfrac{}{}{0px}{}{2n}{n}\right)+[n\text{ even}]\left(\genfrac{}{}{0px}{}{n}{n\mathrm{/}2}\right))$; 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 $n=6,\dots ,5000$.