A158200 A coefficients of characteristic polynomials of binomial modulo two matrices times their doubled transposes: t(n,m,d)=If[ m <= n, Mod[Binomial[n, m], 2], 0]; M(d)=*Transpose[t(n,m,d)].t(n,m,d).Transpose[t(n,m,d)].

1, 1, -1, 1, -2, 1, 1, -7, 7, -1, 1, -8, 14, -8, 1, 1, -13, 46, -46, 13, -1, 1, -18, 99, -196, 99, -18, 1, 1, -37, 279, -763, 763, -279, 37, -1, 1, -38, 316, -1042, 1526, -1042, 316, -38, 1, 1, -43, 486, -2266, 4884, -4884, 2266, -486, 43, -1, 1, -48, 677, -4144, 12122

Offset

0,5

Comments

Row sums are:

{1, 0, 0, 0, 0, 0, -32, 0, 0, 0, 0,...},

Example matrix:

M(3)={{2, 1, 4},

{1, 2, 4},

{1, 1, 3}}

Links

Table of n, a(n) for n=0..59.

Formula

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

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

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

Example

{1},
{1, -1},
{1, -2, 1},
{1, -7, 7, -1},
{1, -8, 14, -8, 1},
{1, -13, 46, -46, 13, -1},
{1, -18, 99, -196, 99, -18, 1},
{1, -37, 279, -763, 763, -279, 37, -1},
{1, -38, 316, -1042, 1526, -1042, 316, -38, 1},
{1, -43, 486, -2266, 4884, -4884, 2266, -486, 43, -1},
{1, -48, 677, -4144, 12122, -17216, 12122, -4144, 677, -48, 1}

Mathematica

Clear[M, T, d, a, x, a0];
T[n_, m_, d_] := If[ m <= n, Mod[Binomial[n, m], 2], 0];
M[d_] := Transpose[Table[T[n, m, d], {n, 1, d}, {m, 1, d}]].Table[T[n, m, d], {n, 1, d}, {m, 1, d}].Transpose[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: A266309 A122021 A303817 * A220602 A166345 A015110

Adjacent sequences: A158197 A158198 A158199 * A158201 A158202 A158203

Keyword

sign,tabl,uned

Author

Roger L. Bagula, Mar 13 2009

Status

approved