A139813 - OEIS

%I #8 Oct 13 2012 14:46:42

%S 1,2,2,2,2,2,2,6,6,2,2,8,6,8,2,2,10,20,20,10,2,2,12,30,20,30,12,2,2,

%T 14,42,70,70,42,14,2,2,16,56,112,70,112,56,16,2,2,18,72,168,252,252,

%U 168,72,18,2,2,20,90,240,420,252,420,240,90,20,2

%N A polynomial triangle based on cross binomial Hodge number matrices/ Hodge diamonds that represent Calabi-Yau type binomials and their monomials.

%C The matrices M(n X n): crossed Pascal matrices:

%C {{1}}

%C ---

%C {{1,1},

%C {1,1}}

%C ---

%C {{1,0,1},

%C {0,2,0},

%C {1,0,1}}

%C ---

%C {{1,0,0,1},

%C {0,3,3,0}.

%C {0,3,3,0},

%C {1,0,0,1}}

%C ---

%C {{1,0,0,0,1},

%C {0,4,0,4.0},

%C {0,0,6,0,0},

%C {0,4,0,4.0},

%C {1,0,0,0,1}}

%C ---

%C {{1,0,0,0,01},

%C {0,5,0,0,5.0},

%C {0,0,10,10,0,0},

%C {0,0,10,10,0,0},

%C {0,5,0,0,5.0},

%C {1,0,0,0,01}}

%C Row sums: {1, 4, 6, 16, 26, 64, 108, 256, 442, 1024, 1796, ...}.

%C Fourier-like visualization of the polynomials:

%C s = Table[ParametricPlot3D[{g[[n]] /. x -> Cos[t], g[[

%C n]] /. x -> Sin[t], n3}, {t, -Pi, Pi}], {n, 1, 10}];

%C Show[s, PlotRange -> All]

%C These are Calabi-Yau type n-fold manifolds as Hodge monomial polynomials.

%C The K3 Hodge number matrix/ diamond is ( 2+20*x+23*x^2 ): M is

%C {{1,0,1},

%C {0,20,0},

%C {1,0,1}}

%C That matrix is this M[3] matrix 3X3 with the central 2 multiplied by a constant 10.

%C This kind of polynomial has been a staple of Calabi-Yau Algebraic Geometry of varieties since the early 90's.

%C The highest n-fold Hodge diamond matrices that I found in the literature that gave me this idea was by Rolf Schimrigk (see links).

%D Christian Meyer, Modular Calabi-Yau threefolds, 2005.

%H Rolf Schimmrigk <a href="http://arXiv.org/abs/hep-th/9405086">Mirror Symmetry and String Vacua from a Special Class of Fano Varieties</a>, arXiv:hep-th/9405087

%F Matrices: T(n,m,d)= If[n - m == 0, Binomial[d, n], If[d - n - m == 0, Binomial[d, m], 0]]; T(n,m,d)->Matrix M(d]); Polynomials in two variables: p(x,y,d)=Sum[Sum[M[d][[k, m]]*x^(k - 1)*y^(m - 1), {m, 1, d + 1}], {k, 1, d + 1}]; Sequence is: a(n,m)_out=Coefficients(p(x,1,d)).

%e {1},

%e {2, 2},

%e {2, 2, 2},

%e {2, 6, 6, 2},

%e {2, 8, 6, 8, 2},

%e {2, 10, 20, 20, 10, 2},

%e {2, 12, 30, 20, 30, 12, 2},

%e {2, 14, 42, 70, 70, 42, 14, 2},

%e {2, 16, 56, 112, 70, 112, 56, 16, 2},

%e {2, 18, 72, 168, 252, 252, 168, 72, 18, 2},

%e {2, 20, 90, 240, 420, 252, 420, 240, 90, 20, 2}

%t Clear[T, M, p, a, g] T[n_, m_, d_] := If[n - m == 0, Binomial[d, n], If[d - n - m == 0, Binomial[d, m], 0]]; M[d_] := Table[T[n, m, d], {n, 0, d}, {m, 0, d}]; p[x_, y_, d_] := Sum[Sum[M[d][[k, m]]*x^(k - 1)*y^(m - 1), {m, 1, d + 1}], {k, 1, d + 1}]; g = Table[ExpandAll[p[x, 1, d]], {d, 1, 10}]; a = Join[{{1}}, Table[CoefficientList[p[x, 1, w], x], {w, 1, 10}]]; Flatten[a] Join[{1}, Table[Apply[Plus, CoefficientList[p[x, 1, w], x]], {w, 1, 10}]];

%K nonn,tabl,uned

%O 1,2

%A _Roger L. Bagula_ and _Gary W. Adamson_, May 23 2008