a binomial sum. --corrected typing
Henry Gould
gould at math.wvu.edu
Wed Apr 21 17:53:48 CEST 2004
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,
767502945, 2987013067, 11641557717, 45429853651, 177490745985,...,
then I offer the following 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 , 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
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