A158188 - OEIS

%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