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A158810 Coefficients of the differentiated row polynomials of the triangular Hadamard matrices of A158800: p(x,n)=If[n less than or equal to 2^m,Sum[H(2^m)[[k]]*x^(1-k),{k,1,n}],If[n greater than m then m+1] 0

%I #2 Oct 12 2012 14:54:56

%S 0,-1,0,-2,-1,-2,3,0,0,0,-4,-1,0,0,-4,5,0,-2,0,-4,0,6,-1,-2,3,-4,5,6,

%T -7,0,0,0,0,0,0,0,-8,-1,0,0,0,0,0,0,-8,9,0,-2,0,0,0,0,0,-8,0,10,-1,-2,

%U 3,0,0,0,0,-8,9,10,-11,0,0,0,-4,0,0,0,-8,0,0,0,12,-1,0,0,-4,5,0,0,-8,9,0,0

%N Coefficients of the differentiated row polynomials of the triangular Hadamard matrices of A158800: p(x,n)=If[n less than or equal to 2^m,Sum[H(2^m)[[k]]*x^(1-k),{k,1,n}],If[n greater than m then m+1]

%C Row sums are:

%C {0, -1, -2, 0, -4, 0, 0, 0, -8, 0, 0, 0, 0, 0, 0, 0,...}.

%C The absolute values of the row sums are:

%C {0, 1, 2, 6, 4, 10, 12, 28, 8, 18, 20, 44, 24, 52, 56, 120,...}.

%C In a quantum Heisenberg matrix mechanics based on the triangular Hadamards

%C where the H(n) behave like wave functions Phi(n), these polynomials

%C are equivalent to the time dependent differentials:

%C Hamiltonian.Phi(n)=-Hbar*I*dPhi(n)/dt

%F Sum of the k-th row polynomial:

%F p(x,n)=If[n>2^m,Sum[H(2^m)[[k]]*x^(1-k),{k,1,n}]];

%F t(n,l)=coefficients(p(x,n),x)

%e {0},

%e {-1},

%e {0, -2},

%e {-1, -2, 3},

%e {0, 0, 0, -4},

%e {-1, 0, 0, -4, 5},

%e {0, -2, 0, -4, 0, 6},

%e {-1, -2, 3, -4, 5, 6, -7},

%e { 0, 0, 0, 0, 0, 0, 0, -8},

%e {-1, 0, 0, 0, 0, 0, 0, -8, 9},

%e {0, -2, 0, 0, 0, 0, 0, -8, 0, 10},

%e {-1, -2, 3, 0, 0, 0, 0, -8, 9, 10, -11},

%e {0, 0, 0, -4, 0, 0, 0, -8, 0, 0, 0, 12},

%e {-1, 0, 0, -4, 5, 0, 0, -8, 9, 0, 0, 12, -13},

%e {0, -2, 0, -4, 0, 6, 0, -8, 0, 10, 0, 12, 0, -14},

%e {-1, -2, 3, -4, 5, 6, -7, -8, 9, 10, -11, 12, -13, -14, 15}

%t Clear[HadamardMatrix];

%t MatrixJoinH[A_, B_] := Transpose[Join[Transpose[A], Transpose[B]]];

%t KroneckerProduct[M_, N_] := Module[{M1, N1, LM, LN, N2},

%t M1 = M;

%t N1 = N;

%t LM = Length[M1];

%t LN = Length[N1];

%t Do[M1[[i, j]] = M1[[i, j]]N1, {i, 1, LM}, {j, 1, LM}];

%t Do[M1[[i, 1]] = MatrixJoinH[M1[[i, 1]], M1[[i, j]]], {j, 2, LM}, {i, 1, LM}];

%t N2 = {};

%t Do[AppendTo[N2, M1[[i, 1]]], {i, 1, LM}];

%t N2 = Flatten[N2];

%t Partition[N2, LM*LN, LM*LN]]

%t HadamardMatrix[2] := {{1, 0}, {1, -1}};

%t HadamardMatrix[n_] := Module[{m}, m = {{1, 0}, {1, -1}}; KroneckerProduct[m, HadamardMatrix[n/2]]];

%t M = HadamardMatrix[16];

%t Table[D[Sum[M[[n]][[m]]*x^(m - 1), {m, 1, n}], {x, 1}], {n, 1, Length[M]}];

%t Table[CoefficientList[D[Sum[ M[[n]][[m]]*x^(m - 1), {m, 1, n}], {x, 1}], x], {n, 1, Length[M]}];

%t Flatten[%]

%Y A158800

%K sign,tabl,uned

%O 0,4

%A _Roger L. Bagula_ and _Gary W. Adamson_, Mar 27 2009

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