A123949 - OEIS

%I #15 Jun 03 2023 06:23:28

%S 1,1,-1,1,-2,1,0,-2,3,-1,1,0,-2,0,1,0,-2,-1,3,1,-1,0,0,-3,6,-2,-2,1,0,

%T 2,-9,15,-11,3,1,-1,1,-4,2,6,-1,-6,-1,2,1,0,-2,7,-1,-11,-3,8,4,-1,-1,

%U 0,0,-3,-6,4,18,-9,-2,-3,0,1,0,0,0,-4,3,19,-29,11,-2,2,1,-1,0,0,0,0,4,0,-25,16,26,-20,-4,2,1,0,0,0,-4,11,7,-63,63,8,-34

%N This sequence needs a definition.

%C n-th level Hadamard matrices for Pascal type binary are inverted and multiplied by the corresponding n th level Gray code Hadamard (some of which matrices have determinant zero): the resulting matrix is processed to give a triangular sequence.

%C Matrices:

%C 1 X 1 {{1}}

%C 2 X 2 {{1, 0}, {0, 1}}

%C 3 X 3 {{1, -1, -1}, {0, 1, 1}, {0, 1, 1}}

%C 4 X 4 {{1, 1, 0, 0}, {0, -1, 0, 0}, {0, -1, 0, 1}, {0, 2, 1, 0}}

%C 5 X 5 {{1, 1, -1, -1, -1}, {0, -1, 0, 0, 0}, {0, -1, 0, 1, 1}, {0, 2, 1, 0, 0}, {0, 0, 1, 1, 1}}

%C 6 X 6 {{1, 1, 0, -1, -1, 0}, {0, -1, -1, 0, 0, -1}, {0, -1, 0, 1, 1, 0}, {0, 2, 1, 0, 0, 1}, {0, 0, 0, 1, 1, 0}. {0, 0, 1, 0, 0, 1}}.

%C They don't get interesting until 4 X 4!

%e Triangle begins

%e   1;

%e   1, -1;

%e   1, -2, 1;

%e   0, -2, 3, -1;

%e   1, 0, -2, 0, 1;

%e   0, -2, -1, 3,1, -1;

%e   0, 0, -3, 6, -2, -2, 1;

%e   0, 2, -9, 15, -11,3, 1, -1;

%e   1, -4, 2, 6, -1, -6, -1, 2, 1;

%e   0, -2,7, -1, -11, -3, 8, 4, -1, -1;

%e   0, 0, -3, -6, 4, 18, -9, -2, -3, 0, 1;

%e Polynomials:

%e   1;

%e   1 -  x;

%e   1 - 2x +  x^2;

%e   0 - 2x + 3x^2 -   x^3;

%e   1 + 0x - 2x^2         +   x^4;

%e   0 - 2x -  x^2 +  3x^3 +   x^4 -  x^5;

%e   0 + 0x + 3x^2 +  6x^3 -  2x^4 - 2x^5 + x^6;

%e   0 + 2x - 9x^2 + 15x^3 - 11x^4 + 3x^5 + x^6 -  x^7;

%e   1 - 4x + 2x^2 +  6x^3 -   x^4 - 6x^5 - x^6 + 2x^7 + x^8;

%t c[i_, k_] := Floor[Mod[i/2^k, 2]];

%t b[i_, k_] := If[c[i, k] == 0 && c[ i, k + 1] == 0, 0, If[c[i, k] == 1 && c[i, k + 1] == 1, 0, 1]];

%t An[d_] := Table[If[Sum[b[n, k]*b[m, k], {k, 0, d - 1}] == 0, 1, 0], {n, 0, d - 1}, {m, 0, d - 1}];

%t Bn[d_] := Table[If[Sum[c[n, k]*c[ m, k], {k, 0, d - 1}] == 0, 1, 0], {n, 0, d - 1}, {m, 0, d - 1}];

%t Xn[d_] := MatrixPower[Bn[d], -1].An[d];

%t a = Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[Xn[d], x], x], {d, 1, 20}]];

%t Flatten[%]

%Y Cf. A122944, A121801, A122947.

%K uned,tabl,sign,less,uned

%O 1,5

%A _Gary W. Adamson_ and _Roger L. Bagula_, Oct 26 2006