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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].

%I A158810 #4 Jul 22 2025 06:23:15
%S A158810 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 A158810 -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 A158810 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 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].
%C A158810 Row sums are:
%C A158810 {0, -1, -2, 0, -4, 0, 0, 0, -8, 0, 0, 0, 0, 0, 0, 0,...}.
%C A158810 The absolute values of the row sums are:
%C A158810 {0, 1, 2, 6, 4, 10, 12, 28, 8, 18, 20, 44, 24, 52, 56, 120,...}.
%C A158810 In a quantum Heisenberg matrix mechanics based on the triangular Hadamards
%C A158810 where the H(n) behave like wave functions Phi(n), these polynomials
%C A158810 are equivalent to the time dependent differentials:
%C A158810 Hamiltonian.Phi(n)=-Hbar*I*dPhi(n)/dt
%F A158810 Sum of the k-th row polynomial:
%F A158810 p(x,n)=If[n>2^m,Sum[H(2^m)[[k]]*x^(1-k),{k,1,n}]];
%F A158810 t(n,l)=coefficients(p(x,n),x)
%e A158810 {0},
%e A158810 {-1},
%e A158810 {0, -2},
%e A158810 {-1, -2, 3},
%e A158810 {0, 0, 0, -4},
%e A158810 {-1, 0, 0, -4, 5},
%e A158810 {0, -2, 0, -4, 0, 6},
%e A158810 {-1, -2, 3, -4, 5, 6, -7},
%e A158810 { 0, 0, 0, 0, 0, 0, 0, -8},
%e A158810 {-1, 0, 0, 0, 0, 0, 0, -8, 9},
%e A158810 {0, -2, 0, 0, 0, 0, 0, -8, 0, 10},
%e A158810 {-1, -2, 3, 0, 0, 0, 0, -8, 9, 10, -11},
%e A158810 {0, 0, 0, -4, 0, 0, 0, -8, 0, 0, 0, 12},
%e A158810 {-1, 0, 0, -4, 5, 0, 0, -8, 9, 0, 0, 12, -13},
%e A158810 {0, -2, 0, -4, 0, 6, 0, -8, 0, 10, 0, 12, 0, -14},
%e A158810 {-1, -2, 3, -4, 5, 6, -7, -8, 9, 10, -11, 12, -13, -14, 15}
%t A158810 Clear[HadamardMatrix];
%t A158810 MatrixJoinH[A_, B_] := Transpose[Join[Transpose[A], Transpose[B]]];
%t A158810 KroneckerProduct[M_, N_] := Module[{M1, N1, LM, LN, N2},
%t A158810 M1 = M;
%t A158810 N1 = N;
%t A158810 LM = Length[M1];
%t A158810 LN = Length[N1];
%t A158810 Do[M1[[i, j]] = M1[[i, j]]N1, {i, 1, LM}, {j, 1, LM}];
%t A158810 Do[M1[[i, 1]] = MatrixJoinH[M1[[i, 1]], M1[[i, j]]], {j, 2, LM}, {i, 1, LM}];
%t A158810 N2 = {};
%t A158810 Do[AppendTo[N2, M1[[i, 1]]], {i, 1, LM}];
%t A158810 N2 = Flatten[N2];
%t A158810 Partition[N2, LM*LN, LM*LN]]
%t A158810 HadamardMatrix[2] := {{1, 0}, {1, -1}};
%t A158810 HadamardMatrix[n_] := Module[{m}, m = {{1, 0}, {1, -1}}; KroneckerProduct[m, HadamardMatrix[n/2]]];
%t A158810 M = HadamardMatrix[16];
%t A158810 Table[D[Sum[M[[n]][[m]]*x^(m - 1), {m, 1, n}], {x, 1}], {n, 1, Length[M]}];
%t A158810 Table[CoefficientList[D[Sum[ M[[n]][[m]]*x^(m - 1), {m, 1, n}], {x, 1}], x], {n, 1, Length[M]}];
%t A158810 Flatten[%]
%Y A158810 A158800
%K A158810 sign,tabl,uned
%O A158810 0,4
%A A158810 _Roger L. Bagula_ and _Gary W. Adamson_, Mar 27 2009