<s>[INST] Calculate the determinant of the matrix. Calculate the determinant.
\begin{bmatrix} C_{n}^{p+n} & C_{n}^{p+n+1} & \dots & C_{n}^{p+2n} \\ 
C_{n}^{p+n+1} & C_{n}^{p+n+2} & \dots & C_{n}^{p+2n+1} \\ 
\vdots & \vdots & \dots & \vdots \\ 
C_{n}^{p+2n} & C_{n}^{p+2n+1} & \dots & C_{n}^{p+3n} \end{bmatrix}
 [/INST] For $n=0$, we have
$$\det(C(p,0)) = 1$$
For $n=1$, we have
$$\det\left(\begin{bmatrix} p+1 & p+2\\ p+2 & p+3\end{bmatrix} \right) = -1$$
For $n=2$, we have
$$\det\left(\begin{bmatrix} \dfrac{(p+1)(p+2)}2 & \dfrac{(p+2)(p+3)}2 & \dfrac{(p+3)(p+4)}2\\ \dfrac{(p+2)(p+3)}2 & \dfrac{(p+3)(p+4)}2 & \dfrac{(p+4)(p+5)}2\\ \dfrac{(p+3)(p+4)}2 & \dfrac{(p+4)(p+5)}2 & \dfrac{(p+5)(p+6)}2 \end{bmatrix} \right) = -1$$
From a small MATLAB script I wrote, it looks like:
$$D(n,p) = \begin{cases} +1 & \text{if }n \equiv 0,3\pmod4\\ -1 & \text{if } n \equiv 1,2 \pmod4\end{cases}$$
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