A158202 Coefficients of polynomials from matrix characteristic polynomials: m(n,m,d)=If[ m <= n, Mod[Binomial[n, m], 2], 0]; M(n)=m(n,m,d).Transpose[m(n,m,d)].Transpose[m(n,m,d)].m(n,m,d)

1, 1, -1, 1, -2, 1, 1, -11, 11, -1, 1, -12, 22, -12, 1, 1, -21, 106, -106, 21, -1, 1, -30, 255, -708, 255, -30, 1, 1, -91, 1065, -3963, 3963, -1065, 91, -1, 1, -92, 1156, -5028, 7926, -5028, 1156, -92, 1, 1, -101, 1880, -12688, 34482, -34482, 12688, -1880, 101

Offset

0,5

Comments

Row sums are:

{1, 0, 0, 0, 0, 0, -256, 0, 0, 0, 0,...}.

Absolute value row sums are:

{1, 2, 4, 24, 48, 256, 1280, 10240, 20480, 98304, 405504,...}.

Matrix example is:

M(3)={{3, 2, 2},

{2, 3, 2},

{6, 6, 5}}.

I've only managed to get three levels of these:

(1,5,5,1),(1,7,7,1),(1,11,11,1)

but they seem stable that you get them with combinations of the binomial

matrix, the Transpose[] and Reverse[].

Plotting them shows that they aren't Sierpinski-Pascal at modulo two.

Links

Table of n, a(n) for n=0..53.

Formula

m(n,m,d)=If[ m <= n, Mod[Binomial[n, m], 2], 0];

M(n)=m(n,m,d).Transpose[m(n,m,d)].Transpose[m(n,m,d)].m(n,m,d);

Out_(n,m)=coefficients(characteristicpolynomial(M(n),x),x)

Example

{1},
{1, -1},
{1, -2, 1},
{1, -11, 11, -1},
{1, -12, 22, -12, 1},
{1, -21, 106, -106, 21, -1},
{ 1, -30, 255, -708, 255, -30, 1},
{1, -91, 1065, -3963, 3963, -1065, 91, -1},
{1, -92, 1156, -5028, 7926, -5028, 1156, -92, 1},
{1, -101, 1880, -12688, 34482, -34482, 12688, -1880, 101, -1},
{1, -110, 2669, -25128, 98706, -152276, 98706, -25128, 2669, -110, 1}

Mathematica

Clear[M, T, d, a, x, a0];
T[n_, m_, d_] := If[ m <= n, Mod[Binomial[n, m], 2], 0];
M[d_] := Table[T[n, m, d], { n, 1, d}, {m, 1, d}].Transpose[Table[T[ n, m, d], {n, 1, d}, { m, 1, d}]].Transpose[Table[T[n, m, d], {n, 1, d}, {m, 1, d}]].Table[ T[n, m, d], {n, 1, d}, {m, 1, d}];
a0 = Table[M[d], {d, 1, 10}];
Table[Det[M[d]], {d, 1, 10}];
Table[CharacteristicPolynomial[M[d], x], {d, 1, 10}];
a = Join[{{1}}, Table[CoefficientList[ Expand[CharacteristicPolynomial[M[n], x]], x], {n, 1, 10}]];
Flatten[a]
Join[{1}, Table[Apply[Plus, CoefficientList[ Expand[CharacteristicPolynomial[M[n], x]], x]], {n, 1, 10}]];

Crossrefs

Sequence in context: A297802 A232266 A234013 * A176307 A197648 A066094

Adjacent sequences: A158199 A158200 A158201 * A158203 A158204 A158205

Keyword

sign,tabl,uned

Author

Roger L. Bagula, Mar 14 2009

Status

approved