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%I #8 May 04 2019 07:09:37
%S 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,
%T 48,-12,1,1,-19,102,-228,228,-102,19,-1,1,-20,121,-330,456,-330,121,
%U -20,1,1,-23,176,-628,1167,-1167,628,-176,23,-1,1,-26,239,-1062,2532,-3368
%N 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)].
%C Row sums are 1, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, ...
%C Example matrix:
%C M(3) = {{1, 0, 1},
%C {0, 1, 1},
%C {1, 1, 3}}
%C The traditional Hadamard self-similar matrix construction is on symbols {1,-1}.
%C When instead the symbols {0,1} are use you get:
%C H(2*n) = {{H(n), H(n)},
%C {H(n), 0 }}
%C which turns out to be a rotated Sierpinski-Pascal modulo two as an n X n matrix.
%C Here the Hadamard transpose product of that construction gives a new set of symmetrical polynomials.
%D F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier/North Holland, 1978, pp. 44-48.
%F t(n,m,d) = If[ m <= n, binomial(n, m) mod 2, 0];
%F M(d) = t(n,m,d).Transpose[t(n,m,d)];
%F a(n,m) = coefficients(characteristicpolynomial(M(n),x),x).
%e {1},
%e {1, -1},
%e {1, -2, 1},
%e {1, -5, 5, -1},
%e {1, -6, 10, -6, 1},
%e {1, -9, 25, -25, 9, -1},
%e {1, -12, 48, -78, 48, -12, 1},
%e {1, -19, 102, -228, 228, -102, 19, -1},
%e {1, -20, 121, -330, 456, -330, 121, -20, 1},
%e {1, -23, 176, -628, 1167, -1167, 628, -176, 23, -1},
%e {1, -26, 239, -1062, 2532, -3368, 2532, -1062, 239, -26, 1}
%t Clear[M, T, d, a, x, a0];
%t T[n_, m_, d_] := If[ m <= n, Mod[Binomial[n, m], 2], 0];
%t 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}]];
%t a0 = Table[M[d], {d, 1, 10}];
%t Table[Det[M[d]], {d, 1, 10}];
%t Table[CharacteristicPolynomial[M[d], x], {d, 1, 10}];
%t a = Join[{{1}}, Table[CoefficientList[Expand[ CharacteristicPolynomial[M[n], x]], x], {n, 1, 10}]];
%t Flatten[a]
%t Join[{1}, Table[Apply[Plus, CoefficientList[Expand[CharacteristicPolynomial[M[n], x]], x]], {n, 1, 10}]];
%K sign,tabl,uned
%O 0,5
%A _Roger L. Bagula_ and _Gary W. Adamson_, Mar 13 2009
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