(Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger and arxiv.org)

Written: April 4, 2021.

We numerically estimate the critical exponents of certain enumeration sequences that naturally generalize the famous Catalan and super-Catalan sequences, and raise deep and original questions about their exact values, and whether they are rational numbers.

Added April 7, 2021: Michael Wallner proved that indeed none of these sequences are P-recursive (in the 3D case) and none of the critical exponents are rational numbers, completely answering (in the 3D case) the deep and original questions raised in the article.

read his insightful Email message

Michael Wallner plans to write it up as a short note for the arxiv. As soon as it is there, I will make the promised donation to the OEIS in his honor.

Added May 27, 2021: Michael Wallner just posted his beautiful artice. A donation to the OEIS, in his honor, has been made.

Maple package

• Capone.txt

Sample Input and Output files for Capone.txt

• If you want to see the first few terms, and estimated asymptotics, and most important, "estimated" critical exponents ( of course they are all -3/2 in the 2d cases) for many 2D ballot sequences

  the input file yields the output file

• If you want to see the first few terms, and estimated asymptotics, and most important, "estimated" critical exponents for many 3D ballot sequences

  the input file yields the output file

• If you want to see the first few terms, and estimated asymptotics, and most important, "estimated" critical exponents for a few 4D ballot sequences

  the input file yields the output file

• If you want to see the first 400 terms of the sequence enumerating walks in the 3D cubic lattice from [0,0,0] to [n,2n,2n] such that it always stays in the region 2x ≥y ≥ z that gives the estimate of -3.73122 for the cricial exponent

  the input file yields the output file

• If you want to see many terms of sequences enumerating, in the 2D square lattice n-step walks that say in A*x ≥ B*y, for A ≤ B ≤ (with gcd(A,B)=1) with positive unit steps suggesting that the critical exponent (to each subsesequence for n mod A+B) the critical exponent is 0

  the input file yields the output file

• If you want to see many terms of sequences enumerating, in the 3D square lattice n-step walks that say in A*x ≥ B*y ≥ C*z for A ≤ B ≤ C (gcd(A,B,C)=1) between with estimates for the cricial exponents

  the input file yields the output file

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