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Linear bonding model using Mathematica

  • To: mathgroup at smc.vnet.net
  • Subject: [mg70318] Linear bonding model using Mathematica
  • From: Roger Bagula <rlbagula at sbcglobal.net>
  • Date: Thu, 12 Oct 2006 05:38:26 -0400 (EDT)

The simplest model is the entirely linear one.
These bondings are like straight chain hydocarbons ( without the hydrogens).
An group  Dynkin diagrams as graphs give linear molecule bonding models:
( permutation type groups)
2by2
{{0, 1},
{1, 0}}
3by3
{{0, 1, 0},
{1, 0, 1},
{0, 1, 0}}
4by4
{{0, 1, 0, 0},
{1, 0, 1, 0},
{0, 1, 0, 1},
{0, 0, 1, 0}}
Characteristic Polynomials:
x^2-1
-x^3+2*x
x^4-3*x+1
Triangular sequence:
{1} (added to complete triangle)
{0, -1},
{-1, 0, 1},
{0, 2, 0, -1},
{1, 0, -3, 0, 1},
{0, -3, 0, 4, 0, -1},
{-1, 0, 6, 0, -5, 0, 1},
{0, 4, 0, -10, 0, 6, 0, -1},
{1, 0, -10,  0, 15, 0, -7, 0, 1},
{0, -5, 0, 20, 0, -21, 0, 8, 0, -1},
{-1, 0, 15, 0,-35, 0, 28, 0, -9, 0, 1}

1,0, -1, -1, 0, 1, 0, 2, 0, -1, 1, 0, -3, 0, 1, 0, -3, 0, 4, 0, -1, -1, 
0, 6,
0, -5, 0, 1, 0, 4, 0, -10, 0, 6, 0, -1, 1, 0, -10, 0, 15, 0, -7, 0, 1, 
0, -5,
0, 20, 0, -21, 0, 8, 0, -1, -1, 0, 15, 0, -35, 0, 28, 0, -9, 0, 1, 0, 6, 0,
-35, 0, 56, 0, -36, 0, 10, 0, -1, 1, 0, -21, 0, 70, 0, -84, 0, 45, 0, 
-11, 0,
1, 0, -7, 0, 56, 0, -126, 0, 120, 0, -55, 0, 12, 0, -1, -1, 0, 28, 0, -126,
0, 210, 0, -165, 0, 66, 0, -13, 0, 1, 0, 8, 0, -84, 0, 252, 0, -330, 0, 
220,
0, -78, 0, 14, 0, -1, 1, 0, -36, 0, 210, 0, -462, 0, 495, 0, -286, 0, 
91, 0,
-15, 0, 1, 0, -9, 0, 120, 0, -462, 0, 792, 0, -715, 0, 364, 0, -105, 0, 16,
0, -1, -1, 0, 45, 0, -330, 0, 924, 0, -1287, 0, 1001, 0, -455, 0, 120, 0,
-17, 0, 1, 0, 10, 0, -165, 0, 792, 0, -1716, 0, 2002, 0, -1365, 0, 560, 0,
-136, 0, 18, 0, -1, 1, 0, -55, 0, 495, 0, -1716, 0, 3003, 0, -3003, 0, 
1820,
0, -680, 0, 153, 0, -19, 0, 1

In triangular sequence terms they are like Hermite/ vibrational type 
polynomials
with alternate zeros in the polynomials.

Mathematica code:
An[d_] := Table[If[ n == m + 1 || n == m - 1, 1, 0], {n, 1, d}, {m, 1, d}]
Table[An[d], {d, 2, 20}]
Table[CharacteristicPolynomial[An[d], x], {d, 2, 20}]
Table[CoefficientList[CharacteristicPolynomial[An[d], x], x], {d, 1, 20}]
Flatten[%]

You can cut and paste that into OEIS. It gives
A049310 <http://www.research.att.com/%7Enjas/sequences/A049310>

> Triangle of coefficients of Chebyshev's S(n,x) := U(n,x/2) polynomials 
> (exponents in increasing order).

It appears Chebyshev's also alternate in power.
They are orthogonal polynomials produced by a simple matrix model.
Roger Bagula


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