A140821 Coefficients of Hodge diamond GCD binomial product 'X' matrices as polynomials: matrix example; M={{2,0,2}. {0,2,0], {2,0,2}: M(d, x, y)= Sum[Sum[If[n == m, GCD[d - 1, m - 1], If[n == d - m + 1, GCD[d - 1, n - 1], 0]]*x^(n - 1)*y^(m - 1), {n, 1, d}], {m, 1, d}] .

2, 2, 4, 2, 4, 6, 6, 6, 6, 8, 8, 12, 8, 8, 10, 10, 20, 20, 10, 10, 12, 12, 60, 60, 60, 12, 12, 14, 14, 42, 70, 70, 42, 14, 14, 16, 16, 112, 112, 280, 112, 112, 16, 16, 18, 18, 72, 504, 252, 252, 504, 72, 18, 18

Offset

1,1

Comments

Row sums are:

{0, 4, 10, 24, 44, 80, 228, 280, 792, 1728}.

Links

Table of n, a(n) for n=1..54.

Formula

M(d, x, y)=Sum[Sum[If[n == m, Binomial[d - 1, m - 1]* GCD[d - 1, m - 1], If[n == d - m + 1, Binomial[d - 1, n - 1] *GCD[d - 1, n - 1], 0]]*x^(n - 1)*y^(m - 1), {n, 1, d}], {m, 1, d}] ; a(n,m)=Coefficients(M(n,x,1)).

Example

{},
{2, 2},
4, 2, 4},
{6, 6, 6, 6},
{8, 8, 12, 8, 8},
{10, 10, 20, 20, 10, 10},
{12, 12, 60, 60, 60, 12, 12},
{14, 14, 42, 70, 70, 42, 14, 14},
16, 16, 112, 112, 280, 112, 112, 16, 16},
{18, 18, 72, 504, 252, 252, 504, 72, 18, 18}

Mathematica

Clear[M, y, x] M[d_, x_, y_] := Sum[Sum[If[n == m, Binomial[d - 1, m - 1]* GCD[d - 1, m - 1], If[n == d - m + 1, Binomial[d - 1, n - 1] *GCD[d - 1, n - 1], 0]]*x^(n - 1)*y^(m - 1), {n, 1, d}], {m, 1, d}]; Table[CoefficientList[M[d, x, 1], x], {d, 1, 10}] Flatten[%] Table[Apply[Plus, CoefficientList[M[d, x, 1], x]], {d, 1, 10}]

Crossrefs

Cf. A140685.

Sequence in context: A091820 A171922 A306743 * A063789 A106264 A278535

Adjacent sequences: A140818 A140819 A140820 * A140822 A140823 A140824

Keyword

nonn,uned,tabl

Author

Roger L. Bagula and Mats Granvik, Jul 16 2008

Status

approved