

A123956


Chebyshev polynomial recursion into matrices back into triangular sequence: p(k, x) = 2*x*p(k  1, x)  p(k  2, x). The absolute values of the coefficients of the two triangular sequences match exactly.


2



1, 1, 1, 1, 2, 2, 1, 3, 4, 4, 1, 4, 8, 8, 8, 1, 5, 12, 20, 16, 16, 1, 6, 18, 32, 48, 32, 32, 1, 7, 24, 56, 80, 112, 64, 64, 1, 8, 32, 80, 160, 192, 256, 128, 128, 1, 9, 40, 120, 240, 432, 448, 576, 256, 256, 1, 10, 50, 160, 400, 672, 1120, 1024, 1280, 512, 512
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OFFSET

1,5


COMMENTS

My reasoning: The Steinbach matrices produce Chebyshev polynomials, which are orthogonal Polynomials.
They are determinantone matrices and if we include Legendre polynomials that give Matrices such that they are elements of SL(n,Q) special linear groups with Q>rational elements that give characteristic polynomials that are orthogonal to each other.
Since the Bonacci and Steinbach are both Fibonacci like, they are associated with tori. The SL[n,C] (complex domain) are associated with the permutation groups A[n].
So the Steinbach matrices are behaving like special kinds of permutation groups (a subset of the larger group: Q < C).
So as long as the determinants are one or negative one and the matrix elements are integers or fractions, they are a special kind of nth level permutation group associated with toral polynomials.
I also figure the connection of the Bessel type polynomials which are Laurent expansion types to the Hermite Taylor expansion: Laurent outside the unit disk:> Bessel> Sum[a[n]/x^n,{n,1,Infinity}] Taylor inside the unit disk:> Hermite (vibrational)> Sum[a[n]*x^n,{n,0,Infinity}].
If we associate the unit disk to determinant one, that even begins to make some sense! I figure since the Chebyshev SL(2,Z) integer group is poster boy for these matrices I'd do that: Polynomial recursion for Chebyshev's is: p[k_, x_] := p[k, x] = 2*x*p[k  1, x]  p[k  2, x] Matrices my program gives: 1 X 1 {{0}}, 2 X 2 {{0, 1}, {1, 0}}, 3 X 3 {{0,1, 0}, {0, 0, 1}, {2, 2, 0}}, 4 X 4 {{0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}, {4, 4, 3, 0}}, 5 X 5 {{0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}, {0, 0, 0, 0, 1}, {8, 8, 8, 4, 0}}, 6 X 6 {{0, 1, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0}, {0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 1}, {16, 16, 20, 12, 5, 0}}.


REFERENCES

CRC Standard Mathematical Tables and Formulae, 16th ed. 1996, p. 484.
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 799.


LINKS

Table of n, a(n) for n=1..66.
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].


FORMULA

p(k, x) = 2*x*p(k  1, x)  p(k  2, x) p(k,x)>a(n,m) a(n,m)>t(n,m)


EXAMPLE

Triangle begins:
{1},
{1, 1},
{1, 2, 2},
{1, 3, 4, 4},
{1, 4, 8, 8, 8},
{1, 5, 12, 20, 16, 16},
{1, 6, 18, 32, 48, 32, 32},
{1, 7, 24, 56, 80, 112,64, 64},
{1, 8, 32, 80, 160, 192, 256, 128, 128},
{1, 9, 40, 120, 240, 432, 448, 576, 256, 256},
{1, 10, 50, 160, 400, 672, 1120, 1024, 1280, 512, 512}
The absolute values of the coefficients of the two triangular sequences match exactly.
Table[Abs[Flatten[b][[n]]]  Abs[Flatten[w][[n]]], {n, 1, Min[Length[Flatten[b]], Length[Flatten[w]]]}]
{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}


MATHEMATICA

p[ 1, x] = 0; p[0, x] = 1; p[1, x] = x + 1; p[k_, x_] := p[k, x] = 2*x*p[k  1, x]  p[k  2, x]; w = Table[CoefficientList[p[n, x], x], {n, 0, 10}] An[d_] := Table[If[n == d && m <n, w[[n]][[d  m + 1]], If[m == n + 1, 1, 0]], {n, 1, d}, {m, 1, d}]; b = Table[CoefficientList[ExpandAll[y^(d  1)*(CharacteristicPolynomial[An[d], x] /. x > 1/y)] /. 1/y > 1, y], {d, 1, 11}]; Flatten[%]


CROSSREFS

Cf. A123235.
Sequence in context: A303273 A193820 A216368 * A113594 A246425 A286468
Adjacent sequences: A123953 A123954 A123955 * A123957 A123958 A123959


KEYWORD

uned,tabl,sign


AUTHOR

Roger L. Bagula and Gary W. Adamson, Oct 27 2006


STATUS

approved



