A140283 A weighted crossed binomial Hodge diamond triangle from coefficients of Hodge polynomials as monomials.

1, 2, 2, 2, 20, 2, 2, 72, 72, 2, 2, 96, 108, 96, 2, 2, 100, 380, 380, 100, 2, 2, 96, 510, 520, 510, 96, 2, 2, 84, 546, 1820, 1820, 546, 84, 2, 2, 80, 560, 2464, 2380, 2464, 560, 80, 2, 2, 72, 504, 2856, 8316, 8316, 2856, 504, 72, 2, 2, 60, 450, 2880, 11340, 10584, 11340

Offset

1,2

Comments

The matrices are:

{1}}.

{{1, 1},

{1, 1}},

{{1, 0, 1},

{0, 20, 0},

{1, 0, 1}},

{{1,0, 0, 1},

{0, 36, 36, 0},

{0, 36, 36, 0},

{1, 0, 0, 1}},

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

{0, 48, 0, 48, 0},

{0, 0, 108, 0, 0},

{0, 48, 0, 48, 0},

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

Row sums are:

{1, 4, 24, 148, 304, 964, 1736, 4904, 8592, 23500, 40048};

These matrices were designed from the K3 like Hodge diamond:

A={{1,0,1},

{0,20,0],

{1,0,1]};

such that weights gave this matrix in a 'smooth' functional pattern.

Links

Table of n, a(n) for n=1..62.

Formula

Weight Function; f(n.d) = Floor[2 + 4*d*Sech[d/2 - n]] Matrix: T(n,m,d)= If[(n == d && m == 0) || (n == 0 && m == d) || (n == 0 && m == 0) || n*m == d^2, 1, If[n - m == 0, f[n, d]* Binomial[d, n], If[d -n - m == 0, f[m, d]*Binomial[d, m], 0]]] Binomial polynomial function: p(x,y,d) := Sum[Sum[M[d][[k, m]]*x^(k - 1)*y^(m - 1), {m, 1, d + 1}], {k, 1, d + 1}] Out_n,m=Coefficients(p(x,1,d)).

Example

{{1},
{2, 2},
{2, 20, 2},
{2, 72, 72, 2},
{2, 96, 108, 96, 2},
{2, 100, 380, 380, 100,2},
{2, 96, 510, 520, 510, 96, 2},
{2, 84, 546, 1820, 1820, 546, 84, 2},
{2, 80, 560,2464, 2380, 2464, 560, 80, 2},
{2, 72, 504, 2856, 8316, 8316,2856, 504, 72, 2},
{2, 60, 450, 2880, 11340, 10584, 11340, 2880, 450,60, 2}}

Mathematica

Clear[T, M, p, a, g, f]; f[n_, d_] = Floor[2 + 4*d*Sech[d/2 - n]]; T[n_, m_, d_] := If[(n == d && m == 0) || (n == 0 && m == d) || (n == 0 && m == 0) || n*m == d^2, 1, If[n - m == 0, f[n, d]* Binomial[d, n], If[d - n - m == 0, f[m, d]*Binomial[d, m], 0]]]; M[d_] := Table[T[n, m, d], {n, 0, d}, {m, 0, d}]; TableForm[Table[M[d], {d, 1, 10}]]; 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: A087238 A226935 A099640 * A067097 A260082 A153438

Adjacent sequences: A140280 A140281 A140282 * A140284 A140285 A140286

Keyword

nonn,uned,tabl

Author

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

Status

approved