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.
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
KEYWORD
AUTHOR
Roger L. Bagula and Gary W. Adamson, Mar 13 2009
STATUS
approved