Proof of Mathar's conjecture on OEIS A027182: a closed form and the six-term recurrence

We prove the recurrence for A027182 stated as a conjecture by R. J. Mathar (OEIS A027182, FORMULA section, 6 May 2026) for all n >= 6, by evaluating the middle coefficient of the square of each A027170 row polynomial. Row symmetry reduces the convolution to five binomial sums, and a shifted second-order recurrence gives the six-term recurrence. The same central-binomial identity proves Mathar's conjectured recurrences for A027171, A027174 and A027175; a parity decomposition also proves his seven-term recurrence for the half-row sums A027179.