<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<HTML><HEAD>
<META http-equiv=Content-Type content="text/html; charset=iso-8859-1">
<META content="MSHTML 6.00.2600.0" name=GENERATOR></HEAD>
<BODY bottomMargin=0 leftMargin=3 topMargin=0 rightMargin=3>
<DIV></DIV>
<DIV>Dear Seqfans, <BR> Copied below (with
permission) is a very interesting letter <BR>I received regarding triangle
A114176: </DIV>
<DIV><A
href="http://www.research.att.com/~njas/sequences/A114176">http://www.research.att.com/~njas/sequences/A114176</A></DIV>
<DIV> </DIV>
<DIV>Would someone be able to verify that the triangle of coefficients
</DIV>
<DIV>needed in the formula
for the amplitude A(N) described below <BR>is indeed given
by triangle <FONT size=2>A114176</FONT>? </DIV>
<DIV> </DIV>
<DIV>If so, it would be a nice example of a how the OEIS is
helpful to researchers. <BR>Thanks, </DIV>
<DIV><FONT size=2> Paul <BR> </FONT></DIV>
<DIV><FONT size=2>PS. One of the example lines of A114176 has an error
-- <BR>I have submitted a correction to the example line to be:
</FONT></DIV>
<DIV>column 2: (1 + 3*x + 1*x^2)/(1-x)^3/(1-x^2)^2 <BR>= 1 + 6*x + 18*x^2 +
43*x^3 + 86*x^4 + 156*x^5 +... </DIV>
<DIV>to avoid confusion. </DIV>
<DIV><FONT
size=2>-----------------------------------------------------------------</FONT></DIV>
<DIV><FONT size=2>On Fri, 16 Jun 2006 16:39:58 -0400 Jocelyn Veilleux <<A
href="mailto:Jocelyn.Veilleux@USherbrooke.ca">Jocelyn.Veilleux@USherbrooke.ca</A>>
writes:</FONT></DIV>
<DIV>Good morning Mr. Hanna,<BR> <BR>I am interested in the coefficients of
your OEIS sequence id:A114176 <BR>for some modeling work I am doing in optics.
<BR>[...]<BR>Now, how do I find this triangle useful... I will try to make a
short story.<BR> <BR>I am working with Optical Coherence Tomography (OCT),
a non-destructive imaging<BR>technique that applies Low-Coherence Interferometry
(LCI) and optical<BR>heterodyne detection to achieve a high axial resolution
(1-10 um) and a high<BR>sensitivity to weakly backscattered light within
translucent materials.<BR>Especially, we are interested in the non-destructive
evaluation of<BR>plasma-sprayed ceramic coatings.<BR> <BR>Since these
coatings have a very complex microstructure, we are doing some<BR>modeling to
better understand the signals. One model considers a light beam<BR>impinging on
a pile of glass plates, each plate being separated from the other<BR>by a thin
film of air. Then, we are interested by the amplitude of the signal<BR>(not the
intensity) being reflected from each interfaces within the plate pile.<BR>In a
first approximation, we replace the two interfaces glass-air and
air-glass<BR>located between two plates by one equivalent interface and both the
equivalent<BR>reflexion (req) and equivalent transmission (teq) coefficients
through this<BR>single interface are calculated accordingly.<BR> <BR>To
calculate the signal amplitude being reflected by the Nth glass plate
(which<BR>corresponds to an optical path z(N)), one must consider all the
contributions<BR>having an optical path z(N) that come from any combination of
reflections<BR>within the first (N-1), (N-2), (N-3)... 3, 2, 1 plates located
over the Nth<BR>plate.<BR> <BR>Here are coming the triangle coefficients.
We expressed an asymptotic value for<BR>the maximum amplitude A of the reflected
signal from an optical path z(N), with<BR>N being the number of plates, in this
way :<BR> <BR>A(N) = t1*t2 * Sum[C(N,i) * teq^(2(i-1)) * req^(N-i+1) *
r2^(N-i), i=1..N],<BR> <BR>where t1 and t2 stand for the transmission
coefficient in amplitude from an<BR>air-glass and a glass-air interface
respectively, r2 is the reflection<BR>coefficient from a glass-air interface,
req and teq were described earlier, and<BR>C(N,i) are the triangle
coefficients.<BR> <BR>I found the coefficients by doing manually the
calculation for N=1,2,3,4,5 and<BR>the beginning part of 6.<BR> <BR>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<BR>you
have done.<BR> <BR>I hope this answers your
question.<BR> <BR>Regards,<BR>Jocelyn Veilleux<BR>Department of
Chemical Engineering<BR>University of Sherbrooke<BR>Sherbrooke (QC),
CANADA</DIV>
<DIV> </DIV>
<DIV><FONT
size=2>-----------------------------------------------------------------</FONT></DIV>
<DIV><FONT size=2>Mon, 19 Jun 2006 11:56:59 -0400</FONT> Jocelyn Veilleux <<A
href="mailto:Jocelyn.Veilleux@USherbrooke.ca">Jocelyn.Veilleux@USherbrooke.ca</A>>
writes:<BR>Dear Paul,<BR> <BR>You have my permission to forward both my
note and e-mail to the group of<BR>professional mathematicians.<BR> <BR>I
am very interested in having some theoritical results that could be applied
to<BR>my model. It will be a pleasure to collaborate with your
group.<BR> <BR>However, please note that I will be out of office from July
5 to August 10.<BR>Therefore, I might not be able to reply your inquiries very
quickly.<BR> <BR>Best regards,<BR>Jocelyn</DIV></BODY></HTML>