A158188 Characteristic polynomials of a binomial modulo two Hadamard transpose general matrix: t(n,m,d) = If[ m <= n, binomial(n, m) mod 2], 0]; M(d)=t(n,m,d).Transpose[t(n,m,d)].

1, 1, -1, 1, -2, 1, 1, -5, 5, -1, 1, -6, 10, -6, 1, 1, -9, 25, -25, 9, -1, 1, -12, 48, -78, 48, -12, 1, 1, -19, 102, -228, 228, -102, 19, -1, 1, -20, 121, -330, 456, -330, 121, -20, 1, 1, -23, 176, -628, 1167, -1167, 628, -176, 23, -1, 1, -26, 239, -1062, 2532, -3368

Offset

0,5

Comments

Row sums are 1, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, ...

Example matrix:

M(3) = {{1, 0, 1},

{0, 1, 1},

{1, 1, 3}}

The traditional Hadamard self-similar matrix construction is on symbols {1,-1}.

When instead the symbols {0,1} are use you get:

H(2*n) = {{H(n), H(n)},

{H(n), 0 }}

which turns out to be a rotated Sierpinski-Pascal modulo two as an n X n matrix.

Here the Hadamard transpose product of that construction gives a new set of symmetrical polynomials.

References

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier/North Holland, 1978, pp. 44-48.

Links

Table of n, a(n) for n=0..60.

Formula

t(n,m,d) = If[ m <= n, binomial(n, m) mod 2, 0];

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

a(n,m) = coefficients(characteristicpolynomial(M(n),x),x).

Example

{1},
{1,  -1},
{1,  -2,   1},
{1,  -5,   5,    -1},
{1,  -6,  10,    -6,    1},
{1,  -9,  25,   -25,    9,    -1},
{1, -12,  48,   -78,   48,   -12,    1},
{1, -19, 102,  -228,  228,  -102,   19,    -1},
{1, -20, 121,  -330,  456,  -330,  121,   -20,   1},
{1, -23, 176,  -628, 1167, -1167,  628,  -176,  23,  -1},
{1, -26, 239, -1062, 2532, -3368, 2532, -1062, 239, -26, 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}]];
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: A058676 A147649 A147644 * A176625 A197342 A197217

Adjacent sequences: A158185 A158186 A158187 * A158189 A158190 A158191

Keyword

sign,tabl,uned

Author

Roger L. Bagula and Gary W. Adamson, Mar 13 2009

Status

approved