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

1,2

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}}

References

Christian Meyer, Modular Calabi-Yau threefolds, 2005.

Links

Table of n, a(n) for n=1..66.

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

Example

{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}

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}]];

Crossrefs

Sequence in context: A105255 A351023 A140818 * A172009 A299150 A202448

Adjacent sequences: A139810 A139811 A139812 * A139814 A139815 A139816

Keyword

nonn,tabl,uned

Author

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

Status

approved