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, -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, -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, 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

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Mar 27 2009

Keywords

• sign
• tabl
• uned

Comments

Row sums are:

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

The absolute values of the row sums are:

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

In a quantum Heisenberg matrix mechanics based on the triangular Hadamards

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

are equivalent to the time dependent differentials:

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

Examples

			{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, -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, 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, 12, -13},
{0, -2, 0, -4, 0, 6, 0, -8, 0, 10, 0, 12, 0, -14},
{-1, -2, 3, -4, 5, 6, -7, -8, 9, 10, -11, 12, -13, -14, 15}
		

Crossrefs

A158800

Programs

Mathematica

Clear[HadamardMatrix];
MatrixJoinH[A_, B_] := Transpose[Join[Transpose[A], Transpose[B]]];
KroneckerProduct[M_, N_] := Module[{M1, N1, LM, LN, N2},
M1 = M;
N1 = N;
LM = Length[M1];
LN = Length[N1];
Do[M1[[i, j]] = M1[[i, j]]N1, {i, 1, LM}, {j, 1, LM}];
Do[M1[[i, 1]] = MatrixJoinH[M1[[i, 1]], M1[[i, j]]], {j, 2, LM}, {i, 1, LM}];
N2 = {};
Do[AppendTo[N2, M1[[i, 1]]], {i, 1, LM}];
N2 = Flatten[N2];
Partition[N2, LM*LN, LM*LN]]
HadamardMatrix[2] := {{1, 0}, {1, -1}};
HadamardMatrix[n_] := Module[{m}, m = {{1, 0}, {1, -1}}; KroneckerProduct[m, HadamardMatrix[n/2]]];
M = HadamardMatrix[16];
Table[D[Sum[M[[n]][[m]]*x^(m - 1), {m, 1, n}], {x, 1}], {n, 1, Length[M]}];
Table[CoefficientList[D[Sum[ M[[n]][[m]]*x^(m - 1), {m, 1, n}], {x, 1}], x], {n, 1, Length[M]}];
Flatten[%]

Formula

Sum of the k-th row polynomial:

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

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