Hi! Rubine of Wildberger & Rubine here. Great review above, thanks!

I don't get the X after Node or the ellipsis here. The typical Catalan grammar, e.g for a binary plane tree, is the simple recursion:

T: Leaf | Node(T,T)

It becomes a multiset equation, then algebra when we count nodes.

Thanks again. As mathematical research goes, this paper isn't overly difficult, so don't be scared of downloading the pdf folks. Much of the paper stems from doing our homework of problems 7.22 and 7.50 in Concrete Mathematics.

Coauthor here. G[n], a single natural number index, is a Catalan number, G[n]=C[n+1]. Similarly, G[0,0,0...,m_k] (with k-2 zeros) is a Fuss number, C[0,0,0,...,1+m_k]. If you mean for an arbitrary vector k, that's still unknown, though I have proven the conjecture about G[0,0,0,...,m_k,m_{k+1}].

Thanks for the shout-out! As math research goes, this isn't overly difficult, so don't be afraid to download the pdf, folks.

www.tandfonline.com/doi/epdf/10....

Thanks for the shout-out. Back when I first learned about them, I thought Catalan referred to the region as well, but no, Eugène Charles Catalan was actually Belgium / French. Catalan's relatively minor contributions was in 1839 and the series wasn't really named after him until Riordan, 1968.

Thanks again. As mathematical research goes, this paper isn't overly difficult, so don't be scared of downloading the pdf folks. Much of the paper stems from doing our homework of problems 7.22 and 7.50 in Concrete Mathematics.

Thanks for the shout-out. I used to complain about the media printing press releases, but now I see whenever they deviate from the press release they get it wrong.

To the math folks out there, Galois Theory of course remains valid; we just used power series instead of radicals so it didn't apply.

Coauthor here. G[n], a single natural number index, is a Catalan number, G[n]=C[n+1]. Similarly, G[0,0,0...,m_k] (with k-2 zeros) is a Fuss number, C[0,0,0,...,1+m_k]. If you mean for an arbitrary vector k, that's still unknown, though I have proven the conjecture about G[0,0,0,...,m_k,m_{k+1}].