The Domb Ap'ery-limit and a proof of the Ramanujan Machine conjecture Z2

We prove that the ratio $B_n\mathrm{/}{D}_n$ of the Apéry-like sequence $B_n$ to the Domb numbers $D_n$ converges to $(7\mathrm{/}24)\zeta (3)$, and that ${\sum }_{n=1}^{\mathrm{\infty }}{64}^n\mathrm{/}(n^3{D}_n{D}_{n-1})=(56\mathrm{/}3)\zeta (3)$. As a corollary we establish the value $Z_2=12\mathrm{/}(7\zeta (3))$ conjectured by the Ramanujan Machine project. The proof uses level-6 eta products, Atkin–Lehner involutions, and Eichler integrals of weight-4 modular forms.