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A123949 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. 1
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, 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, 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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Matrices: 1 X 1 {{1}} 2 X 2 {{1, 0}, {0, 1}} 3 X 3 {{1, -1, -1}, {0, 1, 1}, {0, 1, 1}} 4 X 4 {{1, 1, 0, 0}, {0, -1, 0, 0}, {0, -1, 0, 1}, {0, 2, 1, 0}} 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}} 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}} They don't get interesting until 4 X 4!

LINKS

Table of n, a(n) for n=1..101.

FORMULA

x(i,j)=a(i,j)^(-1).b(i,j) p(n,x)=CharacteristicPolyynomial(x(i,j)) p(n,x)->t(n,m)

EXAMPLE

Triangular sequence:

{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, 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},

{0, 0, -3, -6, 4, 18, -9, -2, -3, 0, 1}

Polynomials:

1,

1 - x,

1 - 2 x + x^2,

0 -2x + 3x^2 - x^3,

1 +0x - 2x^2 + x^4,

0-2x - x^2 + 3 x^3 + x^4 - x^5,

0+0x +3x^2 + 6 x^3 - 2 x^4 - 2 x^5 + x^6,

0+ 2x - 9 x^2 + 15x^3 - 11 x^4 + 3 x^5 + x^6 -x^7,

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

MATHEMATICA

c[i_, k_] := Floor[Mod[i/2^k, 2]]; 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]]; 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}]; 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}]; Xn[d_] := MatrixPower[Bn[d], -1].An[d]; a = Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[Xn[d], x], x], {d, 1, 20}]]; Flatten[%]

CROSSREFS

Cf. A122944, A121801, A122947.

Sequence in context: A131084 A143067 A219605 * A236358 A144082 A145579

Adjacent sequences:  A123946 A123947 A123948 * A123950 A123951 A123952

KEYWORD

uned,tabl,sign

AUTHOR

Gary W. Adamson and Roger L. Bagula, Oct 26 2006

STATUS

approved

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Last modified September 15 14:53 EDT 2019. Contains 327078 sequences. (Running on oeis4.)