

A139360


Coefficients of Chebyshev T(x,n) constructed polynomials that are based on a Cumrun Vafa type of CalabiYau crystal (binomials of this type would behave much like Chladni standing waves): p(x,n,m)=T(x,n)*T(x,m)+T(x,n)+T(x,m): binomials are: p(x,y,n,m)=T(x,n)*T(y,m)+T(x,n)+T(y,m). Integrate(p(x,n,m)/sqrt[1x^2),{x,1,1}]=0 if n,m>0 and n does not equal m: for n=m the result is Pi/2: they are orthogonal polynomials.


1



1, 2, 1, 0, 4, 1, 0, 2, 2, 1, 6, 0, 8, 0, 2, 3, 4, 4, 1, 0, 2, 6, 0, 8, 3, 0, 16, 0, 16, 1, 2, 8, 8, 8, 8, 1, 0, 4, 0, 16, 0, 16, 1, 6, 8, 32, 8, 56, 0, 32, 1, 10, 0, 40, 0, 32, 0, 6, 5, 20, 20, 16, 16, 1, 0, 2, 10, 0, 40, 0, 32, 0, 2, 15, 16, 80, 16, 128, 0, 64, 1, 10, 8, 80, 8, 232, 0, 288, 0, 128, 1, 0, 36, 0, 96, 0
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OFFSET

1,2


COMMENTS

All the row sums are 3.
These polynomials are two level triangles:
m levels and n levels.
The integration table is:
TableForm[Table[Integrate[p[x, n, m]/Sqrt[1  x^2], {x, 1, 1}], {n, 0, 10}, {m, 0, 10}]]
As binomials these polynomials are the quantum mechanics of a type of 2 dimensional crystal that vibrates much like a Chladni standing wave.
They come from thinking of Chebyshev polynomials in terms of a Ring structure in a commutative algebra.


REFERENCES

Dbranes as defects in the CalabiYau crystal. Natalia Saulina, Cumrun Vafa (Harvard U., Phys. Dept.). HUTP04A018, Apr 2004. 28pp. ePrint: hepth/0404246.
Brendan Hassett, Introduction to algebraic Geometry, Cambridge University Press. New York, 2007, p. 237.
Advanced Number Theory, Harvey Cohn, Dover Books, 1963, p. 114.


LINKS

Table of n, a(n) for n=1..96.


FORMULA

p(x,n,m)=T(x,n)*T(x,m)+T(x,n)+T(x,m): For m<n: out_n,m=Coefficients(P(x,n,m).


EXAMPLE

{{1, 2}},
{{1, 0, 4}, {1, 0, 2, 2}},
{{1, 6, 0, 8}, {0, 2, 3, 4, 4}, {1,0, 2, 6, 0, 8}},
{{3, 0, 16, 0, 16}, {1, 2, 8, 8, 8, 8}, {1, 0, 4, 0, 16, 0, 16}, {1, 6, 8, 32, 8, 56, 0, 32}},
{{1, 10, 0, 40, 0, 32}, {0, 6, 5, 20, 20, 16, 16}, {1, 0, 2, 10, 0, 40, 0, 32}, {0, 2, 15, 16, 80, 16, 128, 0, 64}, {1, 10, 8, 80, 8, 232, 0, 288, 0, 128}},
{{1, 0, 36, 0, 96, 0, 64}, {1, 0, 18, 18, 48, 48, 32, 32}, {1, 0, 0, 0, 36, 0, 96, 0, 64}, {1, 0, 18, 54, 48, 216, 32, 288, 0, 128}, {1, 0, 36, 0, 240, 0, 592, 0, 640, 0, 256}, {1, 0, 18,90, 48, 600, 32, 1408, 0, 1408, 0, 512}}


MATHEMATICA

Clear[p, a] p[x_, n_, m_] := ChebyshevT[n, x]*ChebyshevT[m, x] + ChebyshevT[n, x] + ChebyshevT[m, x]; Table[Table[ExpandAll[p[x, n, m]], {m, 0, n  1}], {n, 0, 10}]; a = Table[Table[CoefficientList[p[x, n, m], x], {m, 0, n  1}], {n, 0, 10}]; Flatten[a] Flatten[Table[Table[Apply[Plus, CoefficientList[p[x, n, m], x]], {m, 0, n  1}], {n, 0, 10}]]


CROSSREFS

Sequence in context: A019094 A134082 A185740 * A326759 A140882 A334044
Adjacent sequences: A139357 A139358 A139359 * A139361 A139362 A139363


KEYWORD

tabf,uned,sign


AUTHOR

Roger L. Bagula, Jun 08 2008


STATUS

approved



