The function $\text{sinc}(x)=\frac{\sin x}x$ permeates mathematics and physics in several aspects, and it carries multiple presentations/formulations. My interest is to inject yet another one of such.

Let's understand the determinant $\det(M_{ij})_1^{\infty}$ to mean $\lim_{n\rightarrow\infty}\det(M_{ij})_1^n$. The following has experimental basis, therefore I like to ask:

Question.Is there a proof for the determinantal representation $$\det\left[\frac{(i-1)!}{(2j-1)!}\binom{i^2-\theta^2}j\right]_{i,j=1}^{\infty}=\text{sinc}(\pi\theta) \,\,\,?$$