Integer solutions of Diophantine equation $y^2= 1+4n^{\underline k} $

Question

I am looking for the integer solutions for the diophantine equation $y^2 =4n(n-1)(n-2)\cdots (n-k+1)+1$ for a given $k$ where $n>k+1>5$.

In other words, $$y^2=1+4n^{\underline k},\tag{I}$$ where $n^{\underline k} =n(n-1)(n-2)\cdots(n-k+1)$ is the Pochhammer symbol.

Source of the problem:

The main problem is: Solve the equation $$\frac{m!}{(m-2)!}=\frac{n!}{(n-k)!}$$

$m(m-1)$ must be equal to $n^{\underline k}=\frac{n!}{(n-k)!}$

So we must solve the equation

$$m^2-m -n^{\underline k}=0$$ which has integer solution only if $1+4n^{\underline k}$ is a perfect square, where $n^{\underline k} =n(n-1)(n-2)\cdot\ldots\cdot (n-k+1)$ is the Pochhammer symbol.

Known facts:

1. $n^{\underline k}$ is not a power (see The Product of consecutive integers is never a power by P. Erdos s and J. L. Selfridge), so $4n^{\underline k}$ is not a square.

2. We know, quadratic-residue $\times$ quadratic-nonresidue = quadratic-nonresidue.

3. Steven Stadnicki suggested that there is a link between the problem and Singmaster's conjecture, though I don't understand it.

4. For $k=3$, $E_k (F)$ is an elliptic curve and thus the equation $(I)$ has finite solutions, to be specific $1$. In an elementary way, it can be shown, equation $(I)$ has no solution, since $4P(n,k)+1=(2n^2-6n+2)^2-3$ is a square for finite cases..

5. See the article of Bugeaud, Y.; Mignotte, M.; Siksek, S.; Stoll, M.; Tengely, Sz.: Integral points on hyperelliptic curves. They succesfully treat the equation $$ n(n+1)=\frac{m!}{60(m-5)!} $$ In fact, it follows that $m\in \lbrace{0, 1, 2, 3, 4, 5, 6, 7, 15, 19\rbrace}$, there is hope for equations of type $\binom{n}{\ell}=\binom{m}{k}$.

6. For any $k$ there are finitely many solutions by Siegel's theorem (and for $k>4$ there are finitely many rational solutions by Faltings).

Algebraic geometry:

Let us define a curve $E_k (F):=y^2 =4n(n-1)(n-2)\cdots (n-k+1)+1$ for a given $k$ when $F=\mathbb Z$ or $\mathbb Q$. In other words, $E_k (F):=y^2= 1+4n^{\underline k}$.

Question:

1. Is there infinite integer solutions to the equation $(I)$?
2. Provided the answer to the question is yes, then, what are the integer solutions of the equation $(I)$?

Please consider non-trivial cases.

If above questions are too difficult to answer please, advise how to approach to this problem and what else are known. Also let me know the computational approach in this problem.

EDIT:

One may find relation to Brocard's problem or $abc$ conjecture, but please note that you are refereeing to the problems which are more general in nature, on the other hand, my problem is more specific.