The setup is similar to [this question](https://mathoverflow.net/questions/296056/some-binomial-coefficient-determinants), 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             

```


Some more binomial coefficient determinants
Johann Cigler and I have posted a proof of many of these observations on arXiv:


["An interesting class of Hankel determinants"](https://arxiv.org/abs/1807.08330), arXiv:1807.08330.


Let $d\_r(N)=\det\left({2i+2j+r\choose i+j}\right)\_{i,j=0}^{N-1}$. We show that for $k,n\ge 1$,
\begin{align}
&d\_{2k+1}((2k+1)n)=d\_{2k+1}((2k+1)n+1)=(2n+1)^k,\\
&d\_{2k+1}((2k+1)n+k+1)=(-1)^{k+1\choose 2}4^k(n+1)^k,\\
&d\_{2k}(2kn)=d\_{2k}(2kn+1)=(-1)^{kn},\\
&d\_{2k}(2kn+k)=-d\_{2k}(2kn+k+1)=(-1)^{kn+{k\choose 2}}4^{k-1}(n+1)^{k-1}.
\end{align}


We do not prove the formulas for $d\_r(rn-1)+d\_r(rn+2)$. Possibly these could be proven by modifying the first or last step of the inductions in Lemmas 7.2 or 7.5.


