OEIS Triangle A114176

[Paul D. Hanna]

Dear Seqfans, 
    Copied below (with permission) is a very interesting letter 
I received regarding triangle A114176: 
http://www.research.att.com/~njas/sequences/A114176
 
Would someone be able to verify that the triangle of coefficients 
needed in the formula for the amplitude A(N) described below 
is indeed given by triangle A114176? 
 
If so, it would be a nice example of a how the OEIS is helpful to
researchers. 
Thanks, 
      Paul 
 
PS. One of the example lines of A114176 has an error -- 
I have submitted a correction to the example line to be: 
column 2: (1 + 3*x + 1*x^2)/(1-x)^3/(1-x^2)^2 
= 1 + 6*x + 18*x^2 + 43*x^3 + 86*x^4 + 156*x^5 +... 
to avoid confusion. 
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On Fri, 16 Jun 2006 16:39:58 -0400 Jocelyn Veilleux
<Jocelyn.Veilleux at USherbrooke.ca> writes:
Good morning Mr. Hanna,
 
I am interested in the coefficients of your OEIS sequence id:A114176 
for some modeling work I am doing in optics. 
[...]
Now, how do I find this triangle useful... I will try to make a short
story.
 
I am working with Optical Coherence Tomography (OCT), a non-destructive
imaging
technique that applies Low-Coherence Interferometry (LCI) and optical
heterodyne detection to achieve a high axial resolution (1-10 um) and a
high
sensitivity to weakly backscattered light within translucent materials.
Especially, we are interested in the non-destructive evaluation of
plasma-sprayed ceramic coatings.
 
Since these coatings have a very complex microstructure, we are doing
some
modeling to better understand the signals. One model considers a light
beam
impinging on a pile of glass plates, each plate being separated from the
other
by a thin film of air. Then, we are interested by the amplitude of the
signal
(not the intensity) being reflected from each interfaces within the plate
pile.
In a first approximation, we replace the two interfaces glass-air and
air-glass
located between two plates by one equivalent interface and both the
equivalent
reflexion (req) and equivalent transmission (teq) coefficients through
this
single interface are calculated accordingly.
 
To calculate the signal amplitude being reflected by the Nth glass plate
(which
corresponds to an optical path z(N)), one must consider all the
contributions
having an optical path z(N) that come from any combination of reflections
within the first (N-1), (N-2), (N-3)... 3, 2, 1 plates located over the
Nth
plate.
 
Here are coming the triangle coefficients. We expressed an asymptotic
value for
the maximum amplitude A of the reflected signal from an optical path
z(N), with
N being the number of plates, in this way :
 
A(N) = t1*t2 * Sum[C(N,i) * teq^(2(i-1)) * req^(N-i+1) * r2^(N-i),
i=1..N],
 
where t1 and t2 stand for the transmission coefficient in amplitude from
an
air-glass and a glass-air interface respectively, r2 is the reflection
coefficient from a glass-air interface, req and teq were described
earlier, and
C(N,i) are the triangle coefficients.
 
I found the coefficients by doing manually the calculation for
N=1,2,3,4,5 and
the beginning part of 6.
 
Then I typed 1 1 1 1 3 1 1 6 6 1 1 10 18 10 1 1 15 in Google and found
the work
you have done.
 
I hope this answers your question.
  
Regards,
Jocelyn Veilleux
Department of Chemical Engineering
University of Sherbrooke
Sherbrooke (QC), CANADA
 
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Mon, 19 Jun 2006 11:56:59 -0400 Jocelyn Veilleux
<Jocelyn.Veilleux at USherbrooke.ca> writes:
Dear Paul,
 
You have my permission to forward both my note and e-mail to the group of
professional mathematicians.
 
I am very interested in having some theoritical results that could be
applied to
my model. It will be a pleasure to collaborate with your group.
 
However, please note that I will be out of office from July 5 to August
10.
Therefore, I might not be able to reply your inquiries very quickly.
 
Best regards,
Jocelyn
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