The conjectured recurrence for a(n) holds because the order-5 operator annihilates both the central binomial coefficient and the even-n middle binomial term in the closed form derived from Burnside's lemma.
Fried,Proofs of several conjectures from the OEIS, J
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A short proof of Mathar's 2013 recurrence conjecture for the reversible-binary-string sequence A032123
The conjectured recurrence for a(n) holds because the order-5 operator annihilates both the central binomial coefficient and the even-n middle binomial term in the closed form derived from Burnside's lemma.