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This question was asked on Quora, and I thought I'd try to post a question on MSE to get it answered (because you know, you guys tend to be smarter than them :P)

What is$$\begin{vmatrix}\binom 00 & \binom 10 & \cdots & \binom n0\\\binom 10 & \binom 11 & \cdots & \binom n1\\\vdots & \vdots & \ddots & \vdots\\\binom n0 & \binom n1 & \cdots & \binom nn\end{vmatrix}$$equal to?

If you run the first few values of $n$ in Mathematica, you get a particularly arbitrary-looking sequence$$1,0,-1,8,-71,656,-4816,1920,168784,43920880$$Which doesn't have a particular sequence in oeis. Do you guys have any other insights on this?

I'm also well aware that this problem may not actually have a simplification.

Crescendo
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  • I'm not quite sure I understand the pattern in the coefficients. Could you write out a few more terms and/or write down the general term? – Qiaochu Yuan Sep 25 '17 at 00:29
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    @QiaochuYuan The entries on and below the diagonal are exact to Pascal's triangle. Like,$$\begin{bmatrix}1 & 1 & 1 & 1\\color{brown}1 & \color{brown}1 & 2 & 3\1 & \color{brown}2 & 1 & 3\1 & 3 & 3 & 1\end{bmatrix}$$The two ones (highlighted in brown) add to two, and then you reflect it across the diagonal. Does this help? – Crescendo Sep 25 '17 at 01:29
  • That's helpful, thanks. – Qiaochu Yuan Sep 25 '17 at 03:35

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