A139813 A polynomial triangle based on cross binomial Hodge number matrices/ Hodge diamonds that represent Calabi-Yau type binomials and their monomials.

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, 14, 42, 70, 70, 42, 14, 2, 2, 16, 56, 112, 70, 112, 56, 16, 2, 2, 18, 72, 168, 252, 252, 168, 72, 18, 2, 2, 20, 90, 240, 420, 252, 420, 240, 90, 20, 2

Offset: 1

Author

Roger L. Bagula and Gary W. Adamson, May 23 2008

Keywords

• nonn
• tabl
• uned

Comments

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

{{1}}

---

{{1,1},

{1,1}}

---

{{1,0,1},

{0,2,0},

{1,0,1}}

---

{{1,0,0,1},

{0,3,3,0}.

{0,3,3,0},

{1,0,0,1}}

---

{{1,0,0,0,1},

{0,4,0,4,0},

{0,0,6,0,0},

{0,4,0,4,0},

{1,0,0,0,1}}

---

{{1,0,0,0,0,1},

{0,5,0,0,5.0},

{0,0,10,10,0,0},

{0,0,10,10,0,0},

{0,5,0,0,5,0},

{1,0,0,0,0,1}}

Examples

			{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, 14, 42, 70, 70, 42, 14, 2},
{2, 16, 56, 112, 70, 112, 56, 16, 2},
{2, 18, 72, 168, 252, 252, 168, 72, 18, 2},
{2, 20, 90, 240, 420, 252, 420, 240, 90, 20, 2}
		

References

• Christian Meyer, Modular Calabi-Yau threefolds, 2005.

Links

• Rolf Schimmrigk Mirror Symmetry and String Vacua from a Special Class of Fano Varieties, arXiv:hep-th/9405086, 1994.

Programs

Mathematica

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}]];