%I #6 Nov 04 2013 02:21:54
%S 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,
%T 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,
%U 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
%N 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.
%C All the row sums are 3.
%C These polynomials are two level triangles:
%C m levels and n levels.
%C The integration table is:
%C TableForm[Table[Integrate[p[x, n, m]/Sqrt[1  x^2], {x, 1, 1}], {n, 0, 10}, {m, 0, 10}]]
%C As binomials these polynomials are the quantum mechanics of a type of 2 dimensional crystal that vibrates much like a Chladni standing wave.
%C They come from thinking of Chebyshev polynomials in terms of a Ring structure in a commutative algebra.
%D Dbranes as defects in the CalabiYau crystal. Natalia Saulina, Cumrun Vafa (Harvard U., Phys. Dept.). HUTP04A018, Apr 2004. 28pp. ePrint: hepth/0404246.
%D Brendan Hassett, Introduction to algebraic Geometry, Cambridge University Press. New York, 2007, p. 237.
%D Advanced Number Theory, Harvey Cohn, Dover Books, 1963, p. 114.
%F 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).
%e {{1, 2}},
%e {{1, 0, 4}, {1, 0, 2, 2}},
%e {{1, 6, 0, 8}, {0, 2, 3, 4, 4}, {1,0, 2, 6, 0, 8}},
%e {{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}},
%e {{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}},
%e {{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}}
%t 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}]]
%K tabf,uned,sign
%O 1,2
%A _Roger L. Bagula_, Jun 08 2008
