@Nasser this was enjoyable to read

Wish I had more to say but I'm still only on question 4 so far. I wish the Q, A, and comments were all together for each number, because I am reading the answers one at a time, reading the commentary, and then I have to go back to re-read what the last question was...but I probably need to loosen up because this is just a fun read

@ydd Yes, I agree, Knuth should have put each question/answer in same "block". But if you reach the very end of the article where Knuth sends Wolfram a question about programing something in Mathematica. Wolfram replies to him. Then Knuth uses ChatGPT to ask it to program an answer and he thinks it did better than Wolfram's :)

It is nice to be able to email Wolfram directly a question about coding something in Mathematica and get reply back from him. I guess only if you are knuth you could do this :)

@Nasser I'm inclined to disagree with Knuth there as I think Stephen's answer is more flexible and expression generation via strings is almost always a bad idea but the exchange is fun

For most practical purposes if you know the domain expert, you can email someone at WRI and they'll respond 90% of the time, back in 2019 I got a lot of help with various FE optimizations from John Fultz + gave him a bunch of feedback on what power users might want to do with the system

It woud be interesting to see S.W.'s response about Binomial[-1, -1]. Did WR pick the wrong generalization for negative numbers or was Knuth simply a bit hyperoblic when he said that anyone who has worked on this in the last 50 years knows that Binomial[-1, -1] should be zero and not one like in Mathematica.

@C.E. a thing I appreciate about the Mathematica choice is that the matrix of binomials Array[Binomial, {10, 10}, 0] has an inverse given by the negative binomials Array[Binomial[-#2-1, -#-1]&, {10, 10}, 0]

The positive and negative binomials being inverses of each other seems entirely natural to me

I see. Good observation.

Less an observation and more the fact that oeis.org/A130595 popped up in my research