Dear Doron,

I have seen your paper "Some Deep and Original Questions about the
"critical exponents" of Generalized Ballot Sequences" online today and I
could not resist to take a deeper look. I can tell you that all your
estimated values on page 5 fit quite nicely and all of them (except for
(1,1,1), obviously) are irrational and of the form -1-Pi/arccos(-c) for
an algebraic number c.

Before going into the details, let me comment on your notation (a,b,c):
I am afraid there is an inconsistency comparing your definition on the
top of page 5 and the output in "oCapone2.txt" (and this is also
reflected in the list on page 5). For example, in "oCapone2.txt" you
compute (1,2,2) which you then state as (2,1,1) in the paper. And this
is the case for all of them, but it is consistent. So if you give
(a,b,c) in the list on page 5 you mean (M/c,M/b,M/a) in your original
definition. (Note that swapping a and c gives the same enumeration, as
you simply reverse the direction of the walks.)

Now, the theory to prove the critical exponent comes from the
enumeration of lattice paths in the quarter plane (1st quadrant). I am
sure you know the necessary results, but let me show you how to compute
it on the example of walks from (0,0,0) to (n, 2n, 2n) where you
conjecture 3.731220575. Here we have the constraints 2x >= y >= z

1) We transform the 3D path into a 2D path in the quarter plane by
mapping (x,y,z) to (2x-y, y-z), which follows simply from the
constraints on the domain. This gives a walk from (0,0) to (0,0)
constraint to stay in the quarter plane with steps (2,0), (-1,1),
(0,-1). (Simply map the allowed steps (1,0,0), (0,1,0), and (0,0,1)
using the above mapping.)

2) This is a walk with large steps (not in the set of small steps
{-1,0,1}^2), which was recently treated in "Counting walks with large
steps in an orthant" by Bostan, Bousquet-Mélou, and Melczer; see
https://doi.org/10.4171/jems/1053 and particularly Section 8. The
necessary computations have been implemented in Maple (see
https://www.labri.fr/perso/bousquet/publis.html) and I adapted the
original worksheet "Exponent Filter.mw", which I append. The necessary
code is summarized in Section 3.
So next, we use the "minPs" command and get that the critical exponent
is equal to
-3.7311974809001042931913645385851546712463534092882707004498104372574495582842455
which is
-1 - Pi/arccos(1/sqrt(6)).

3) Next, we prove that this value is irrational using the theory from
"Non-D-finite excursions in the quarter plane" from Bostan, Raschel,
and  Salvy; see https://doi.org/10.1016/j.jcta.2013.09.005. This also
immediately proves that the generating function is not D-finite (see
Theorem 3 therein). Here, we show that a transformed minimal polynomial
of the algebraic number c=1/sqrt(6) is not divisible by any cyclotomic
polynomial (see Section 2.4.2 for details).

That's it. I have checked the others in your list too; all of them are
irrational and therefore their GFs are also not D-finite.

Remark: By design, the walks in the QP always have the steps (A,0),
(0,-B), (-C,C) for positive integers A,B,C. For A=B=C=1 these walks are
called tandem walks (or actually their reflections along y=x).

Best wishes,

Michael Wallner,  Math Dept., University of Vienna, Austria