a binomial sum.

[Henry Gould]

Dear Francois, or should I stay with 'pin'?

I agree with Marc that maybe a comment on Sequence A072547 might be a good
idea.

What I want to remark here is that if we define  f(n) = (-1)^n a(n), so that
we have the sequence of all positive numbers 1, 1, 7, 21, 81, 295, 1107,
4165, 15793, 60171, 230253, 884235, 3406105, 13154947, 50922987, 197519941,
767502 very neat recurrence  relation
 2f(n) + f(n-1) = (3n+1)C(n) + (-1)^n,
where, of course, C(n) = (2n+1)!/n!(n+1)!  is the usual Catalan number. With
f(n)  given this way then we have a sequence of positive numbers with a
tie-in to our good old Catalan numbers.
You can translate the generating function for Sequence A072547 over to one
for f(n) easily.

Now, let's look at the ratio  f(n+1)/f(n). For successive values of n, I
found (approximately)
3,   3.85714,   3.64197,  3.75254,  3.76421,  3.79183,  3.80998,  3.82664,
3.84027,  3.85220, 3.86216,  3.87101,  3.87879,   3.88569,  3.89185,
3.89739, 3.89110,  3.90691, . . . ,
so that we have to ask whether the limit  of  f(n+1)/f(n) exists as  n goes
to ƒ, and if it is less than 4, and indeed does it have 4 as a limit? Maybe
it just gradually diverges?
I just thought of this and haven't even tried to prove anything about it
yet.

A bientot,

Henri