|
%I #5 Oct 13 2012 14:42:08
%S 9,5,4,1,0,8,5,-12,0,16,9,0,-32,0,32,5,20,0,-80,0,64,1,0,72,0,-192,0,
%T 128,5,-28,0,224,0,-448,0,256,9,0,-128,0,640,0,-1024,0,512,5,36,0,
%U -480,0,1728,0,-2304,0,1024,1,0,200,0,-1600,0,4480,0,-5120,0,2048
%N A triangle of coefficients based on A139360 as an n-like set of three binomials: f(x,y,n)=ChebyshevT[n, x]*ChebyshevT[n, y] + ChebyshevT[n, x] + ChebyshevT[n, y]; p(x,y,z,n)=f(x,y,n)+f(y,z,n)+f(z,x,n).
%C The row sums are all 9.
%C The octic Algebraic Variety as an Implicit curve:
%C f[x_, y_, z_] = 9 - 32 x^2 +
%C 32 x^4 - 32 y^2 + 64 x^2 y^2 - 64 x^4 y^2 + 32 y^4 - 64 x^2 y^4 + 64 x^4 y^4 - 32 z^2 + 64 x^2 z^2 - 64 x^4 z^2 + 64 y^2 z^2 - 64 y^4 z^2 + 32 z^4 - 64 x^2 z^4 + 64 x^4z^4 - 64 y^2 z^4 + 64 y^4 z^4 - 1;
%C has a 24 horn structure with an octahedral shaped ellipsoid embedded.
%C These quantum states could be expanded to the full
%C {x,y,z,n,m} to give an analog of a 3d quantum Vafa-Calabi-Yau crystal.
%F f(x,y,n)=ChebyshevT[n, x]*ChebyshevT[n, y] + ChebyshevT[n, x] + ChebyshevT[n, y]; p(x,y,z,n)=f(x,y,n)+f(y,z,n)+f(z,x,n); Out_n,m=Coefficients(P(x,1,1,n)).
%e {9},
%e {5, 4},
%e {1, 0, 8},
%e {5, -12, 0, 16},
%e {9, 0, -32, 0, 32},
%e {5, 20, 0, -80, 0, 64},
%e {1, 0, 72, 0, -192, 0, 128},
%e {5, -28, 0, 224, 0, -448, 0,256},
%e {9, 0, -128, 0, 640, 0, -1024, 0, 512},
%e {5, 36,0, -480, 0, 1728, 0, -2304, 0, 1024},
%e {1, 0, 200, 0, -1600, 0, 4480, 0, -5120, 0, 2048}
%t f[x_, y_, n_] := ChebyshevT[n, x]*ChebyshevT[n, y] + ChebyshevT[n, x] + ChebyshevT[n, y]; Table[ExpandAll[f[x, y, n] + f[y, z, n] + f[x, z, n]], {n, 0, 10}]; a = Table[CoefficientList[ExpandAll[f[x, y, n] + f[y, z, n] + f[x, z, n]] /. y -> 1 /. z -> 1, x], {n, 0, 10}]; Flatten[a]
%Y Cf. A139360.
%K uned,sign
%O 1,1
%A _Roger L. Bagula_ and _Gary W. Adamson_, Jun 12 2008
|
