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 A139360 Coefficients of Chebyshev T(x,n) constructed polynomials that are based on a Cumrun Vafa type of Calabi-Yau 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[1-x^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 (list; graph; refs; listen; history; text; internal format)
 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 D-branes as defects in the Calabi-Yau crystal. Natalia Saulina, Cumrun Vafa (Harvard U., Phys. Dept.). HUTP-04-A018, Apr 2004. 28pp. e-Print: hep-th/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 FORMULA p(x,n,m)=T(x,n)*T(x,m)+T(x,n)+T(x,m): For m

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Last modified December 8 11:36 EST 2022. Contains 358693 sequences. (Running on oeis4.)