A383440 and A383453

[Sela Fried]

As far as I understand there are some inaccuracies (and developments).

First, A383453 is defined to be 

"The Geode Bi-Tri infinite rectangular array, read by upward antidiagonals." 

According to the paper 

N. J. Wildberger and Dean Rubine, A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode, Amer. Math. Monthly (2025), 

the stated formula 

 "The entry (m2, m3) of the rectangular array is equal to (2*m2 + 3*m3 + 3)!/((2*m2 + 2*m3 + 3)*(m2 + m3 + 1)*(m2 + 2*m3 + 2)! * m2! * m3!), with m2, m3 >= 0.

is only conjectural. On the other hand, in A383440, defined to be 

"a(n) = (5*n + 3)!/((8*n^2 + 10*n + 3)*(n!)^2*(3*n + 2)!)."

It is stated in the formula that

a(n) = A383453(2*n, n), conjectured by Wildberger-Rubine to be the main diagonal of the Geode Bi-Tri array G[m_2, m_3].

First, "2*n" should be "n".

Second, the latter should not be stated as a conjecture, since it is just the diagonal of the array. The conjecture should appear in A383453 and is concerned with the formula for the whole array.

In any case, the conjecture was recently settled by TEWODROS AMDEBERHAN AND DORON ZEILBERGER here.

Sela.