Recurrence formulation of the alpha-family Hankel-transform conjecture

Let $a_n(\beta)$ be defined by

$$a_n=a_{n-1}+3a_{n-2}+\sum_{k=1}^{n-3}a_k a_{n-k-2},$$

with $a_0=1$, $a_1=0$, and $a_2=\beta$, and let $h_n(\beta)$ be its Hankel transform. Recurrence-form Hankel-transform conjecture. The Hankel transform is

$$h_n(\beta)=\beta^{\left\lfloor\frac{(n+1)^2}{4}\right\rfloor}[x^n]\frac{(1+x)(1-\beta x^2)}{1-3x^2+\beta x^4}.$$

This is a reformulation of the preceding alpha-family conjecture under $\beta=\alpha-2$.