The setup is similar to this question, but generalizes the size of the Hankel matrix. We'll define

$$d(n,k,r):=\det\left(\binom{2i+2j+k+r}{i+j}\right)_{i,j=0}^{kn-1}.$$
*Edit: Thanks to Johann Cigler for checking, which made me discover that the originally given formula $d(n,k,r):=\det\left(\binom{2i+2j+k}{i+j}\right)_{i,j=0}^{kn-1+r}$ was not the one I used in my implementation. Sorry for that! So in fact my generalization only concerns matrix sizes that are multiples of $k$. Now I am not sure if the picture becomes more coherent when including other matrix sizes as well, but it may be wotrh giving it a look.*

The original question is the special case when $k$ is odd and $r=0$ and conjectures $d(n,k,0)=(2n+1)^\frac{k-1}{2}$. (For even $k$, we have $d(n,k,0)=(-1)^{ n\frac k2}$ according to Krattenthaler.)

Computer experiments suggest that in the more general case, there is still a certain proportion of those determinants which are, up to sign, exact $\bigl[\frac{k+r-1}2\bigr]^\text{th}$ powers (including $\pm1$), while the others are not only no powers but usually have quite large prime factors (e.g. see the two grey entries in the $k=4$ column just below). In the following I will ignore those other values and only consider those which are powers. It may not be a surprise that they all come in intriguing patterns.

E.g. for $r=1$, the table of $d(n,k,1)$ starts with $$\begin{matrix} n \backslash k &2&3&4&5&6&7&8&9&10&11 \\ \hline \ 0 \quad| &1 &1 &1 &1 &1 &1 &1 &1 &1 &1\\ \ 1 \quad|&-4 & 8 & \color{grey} {\scriptstyle {(-61)}} & & & & & \\ \ 2 \quad| &3& -16& -8^2 & -24^2 & & & & \\ \ 3 \quad|&\color{red}5 &1 & \color{grey}{\scriptstyle {(5\cdot139)}} & -36^2 & 12^3 & -48^3 & & \\ \ 4 \quad|&\color{red}{-12} & 1 & 7^2 & & & 64^3 & 16^4 & 80^4 \\ \ 5 \quad|&\color{red}7 &32 & \color{blue}{9^2} & 1 & & & & 100^4& -20^5 & 120^5 \\ \ 6\quad|&9 &-40 & & -1 & 11^3 & & & & & -144^5 \\ \ 7 \quad|&-20 &1 & \color{blue}{-24^2} & & \color{red}{13^3} & 1 & & \\ \ 8 \quad|&11 &1 & & 84^2 & & 1 & 15^4 & \\ \ 9 \quad|&13 & 56 & \color{blue}{15^2} & 96^2 & & & 17^4 & 1 \\ 10\quad|&-28 & -64& 17^2& & \color{red}{36^3} & & & -1 & 19^5 \\ 11\quad|&15 & 1 & & & & -160^3 & & & 21^5 & 1 & \\ 12 \quad|&17 & 1 & -40^2 & & & 176^3 & & & & 1 & \\ 13 \quad|&-36 & 80 & & & \color{red}{23^3} & & 48^4 & \\ \end{matrix} $$

These first entries largely suffice to conjecture the general patterns of $d(n,k,1)$ for even and odd $k$.

For $r=2,3,...$ the patterns are not too different from the ones encountered for $r=1$. (See at the end.)

Instead of $d(n,k,r)$, wherever it is up to sign such a $\bigl[\frac{k+r-1}2\bigr]^\text{th}$ power, let us consider the root $e(n,k,r)$ defined by $$e(n,k,r):=\sqrt[{\bigl[\frac{k+r-1}2 \bigr]}]{|d(n,k,r)|} \in\mathbb N$$

If $r$ and $k$ are fixed, the $e(n+\lambda(k+r),k,r)_{\lambda\in\mathbb N_0}$ always seem to form an arithmetic sequence. For $k+r$ even, its difference is either $2k(r+k)$ or $0$ (in the latter case, we have the constant sequence $1,1,...$), i.e. the first conjecture will be

$$e(n+k+r,k,r)-e(n,k,r)=\begin{cases}2k(k+r)\\ 0\end{cases}.$$

It seems much harder to formalize *where* the $\pm1$'s occur (and where the others), at least for odd $r\ge3$. OTOH, the pattern seems clear if $r$ is even:

For even $r$, we have $|d(n,k,r)|=1$ iff $k$ is also even and $n=\lambda\frac{k+r}{\gcd(k,r)}$, where $\lambda \in\mathbb N_0$. In this case, more precisely, $$d(n,k,r)= (-1)^{ n\frac k2(\frac r2-1)}.$$

Further, if $k$ and $r$ are coprime and moreover $(k+r)$ is odd, **the $k^\text{th}$ column consists of equidistant triples** (three of them are colored in the above table) **such that the middle value of the root is the sum of the two others**, e.g. from $d(10,4,1)$ and $d(12,4,1)$ we can guess $d(14,4,1)=23^2$. BTW, the middle value is always $\equiv0\pmod4$. So for these columns, the roots come in exactly $3$ arithmetic sequences.

There must be a deeper reason for this "sum behavior". Any idea what is happening here?

The density (i.e. the proportion of $\bigl[\frac{k+r-1}2\bigr]^\text{th}$ powers among all values) of a given column is for coprime $k,r$ $$ \begin{cases}\frac3{k+r} & \text{if}\ k\not\equiv r\pmod 2\\ \frac4{k+r} & \text{if} \ k\equiv r\pmod 2\text{ i.e. both odd}\end{cases}$$ and a certain multiple of that if $k,r$ are not coprime.

The first entries of $e(n,k,r)$ for $r=2,...,8$ are displayed below, with a "$-$" sign meaning that the corresponding power $d(n,k,r)$ bears a negative sign. (So in fact, compared with the $d(n,k,1)$ table above I have just omitted the exponents $\bigl[\frac{k+r-1}2\bigr]$ for better lisibility.) Sadly I couldn't get the entries of the columns to flush right, and I did *not* want to do that manually.

```
r=2:
n\k 0 1 2 3 4 5 6 7 8 9 10
0 1 1 1 1 1 1 1 1 1 1 1
1 1 1 -8 -4 12
2 1 -4 1 3 32 8 40
3 1 3 -16 1 5 -72
4 1 3 1 36 1 7
5 1 -8 -24 7 16 1 9
6 1 5 1 -16 1 80 1
7 1 5 -32 9 60 11 120
8 1 -12 1 1 -28
9 1 7 -40 1 15 -192
10 1 7 1 13 84 15 128 1
11 1 -16 -48 -28 36 19
12 1 9 1 15 1 36 1 200 1
13 1 9 -56 108 21
r=3:
n\k 0 1 2 3 4 5 6 7 8 9
0 1 1 1 1 1 1 1 1 1 1
1 1 1 -12 4 -16
2 1 -8 -1 3 -48
3 1 8 3 24 5 1
4 1 1 -8 1 48 1
5 1 1 5 -36 1 -80
6 1 -16 -1 9 -120
7 1 16 48 9 11
8 1 1 7 1 20 1 120 1
9 1 1 -16 -60 11 -96 13 -28
10 1 -24 9 -1 -1 -192
11 1 24 72 17
12 1 1 1 128 17 1
13 1 1 11 -84 1 -1
14 1 -32 -24 -1 17 -264
15 1 32 13 96 36 21 220
r=4:
n\k 0 1 2 3 4 5 6 7 8 9
0 1 1 1 1 1 1 1 1 1 1
1 1 1 16 4 -20
2 1 12 1 3
3 1 -4 -1 32 1 5
4 1 1 -12
5 1 3 -24 5 48 -1
6 1 3 1 12 1 80 1
7 1 7 64
8 1 -8 36 1 11 -24
9 1 -1 80 11 1
10 1 5 1 24 1
11 1 5 -48 96 13 -140 15
12 1 1 11 1 1
13 1 -12 24 112 19
r=5:
n\k 0 1 2 3 4 5 6 7 8 9
0 1 1 1 1 1 1 1 1 1 1
1 1 1 20 -4 24
2 1 4 -1 3
3 1 -12 1 -40
4 1 12 3 32 1 84
5 1 60 7
6 1 -1 -1 -96
7 1 -1 5 -48 7 -80 1 -140
8 1 1 16 1
9 1 24 12 9 100 -24
10 1 -24 -1
11 1 7 1 -120 13 -1
12 1 1 80 1 -28 1
13 1 1 140 15 192 17
r=6:
n\k 0 1 2 3 4 5 6 7 8 9 10
0 1 1 1 1 1 1 1 1 1 1 1
1 1 1 -24 -4 28
2 1 16 1 3
3 1 3 -48
4 1 4 1 40 1 96
5 1 1 -72 7
6 1 32 5 1
7 1 3 -96 1
8 1 3 1 1 140 1
9 1 7 80 9 -120
10 1 48 1 -20 1 13
11 1 8 11 -144
12 1 1 9 1 256
13 1 -168 15
r=7:
n\k 0 1 2 3 4 5 6 7 8 9 10 11
0 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 -28 4 -32
2 1 -20 -1 3
3 1 3 56
4 1 16 1
5 1 -16 3 -40 1 -84 -1
6 1 -72 -1 16
7 1 8 1 -12 112
8 1 1 1 160
9 1 1 5 -140 1 -216
10 1 -1 -1
11 1 9 120 11 168
12 1 32 80 1 -24 1 15
13 1 -32 13 -196
r=8:
n\k 0 1 2 3 4 5 6 7 8 9 10 11 12
0 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 32 4 -36
2 1 -4 1 3
3 1 -20 1 64
4 1 3 -8 1
5 1 4 -1 96 1
6 1 1 84 1
7 1 -1 128 9
8 1 40 7 1
9 1 3 1 160 -1
10 1 3 1 1 216 1
11 1 7 192
12 1 1 1
13 1 -60 -16 11 -168 13 224 -32
```