Hankel determinant identities for convolution powers divisible by three

Establish the conjectured formulas for the Hankel determinants H_n(M(x)^r) of the r-th convolution power of the Motzkin generating function M(x), for every positive integer r divisible by 3: prove that H_{rn}(M(x)^r)=H_{rn+1}(M(x)^r)=(-1)^n(n+1)^{r-1}, and that H_{rn+2}(M(x)^r)+H_{rn-1}(M(x)^r)=32a(n+1)^{r-1}, where |a|=1/[r(r-3)].

Background

The paper defines F(x,r)=M(x)^r, where M(x) is the generating function of the Motzkin numbers, and studies the Hankel determinants H_n(F(x,r)). Explicit formulas are proved for several small convolution powers, including r=3 and r=6.

Conjecture 3 proposes a uniform pattern for all r divisible by 3. The paper later states that the conjectures are verified computationally for r<27, so the unresolved aspect is the extension of these identities to all positive multiples of three beyond the verified range.