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Answer to the title: of course we can interpolate Pascal's triangle with $z(x,y)= {x \choose y}$, but my laptop cannot handle the graph of $z(x,y)$, and I cannot find one online.

I am looking for high-quality visualizations of the interpolation of Pascal's triangle, particularly for $x<0$. Is the triangle still symmetric?

enter image description here

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    $\begingroup$ I took the liberty of adding a plot of $\Gamma(x+y+1)/(\Gamma(x+1)\Gamma(y+1))$, which is the continuous analogue of $\binom{x+y}x=\binom{x+y}y$. Can't avoid the infinities coming from $\Gamma[z]$ having a pole at all the non-positive integer points. This shows in the plot as those spikes and white lines covering points where the function is undefined. Probably marring your viewing pleasure, I'm afraid. $\endgroup$ – Jyrki Lahtonen Feb 20 at 5:30
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    $\begingroup$ Using the beta-function instead may or may not help a bit. $\endgroup$ – Jyrki Lahtonen Feb 20 at 5:50

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