cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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].

Original entry on oeis.org

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

Views

Author

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

Keywords

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)

A173814 Coefficients of Hadamard Cartan G_2 self-similar 2^n matrices:M={{2, -1}, {-3, 2}}.

Original entry on oeis.org

1, 1, -4, 1, 1, -16, 30, -16, 1, 1, -64, 676, -2752, 4678, -2752, 676, -64, 1, 1, -256, 13560, -316160, 3830300, -25002240, 87841480, -180202240, 227671110, -180202240, 87841480, -25002240, 3830300, -316160, 13560, -256, 1, 1, -1024
Offset: 0

Views

Author

Roger L. Bagula, Feb 25 2010

Keywords

Comments

Row sums are:
{1, -2, 0, 400, 0, 231040000000000, 0,...}.

Examples

			{1},
{1, -4, 1},
{1, -16, 30, -16, 1},
{1, -64, 676, -2752, 4678, -2752, 676, -64, 1},
{1, -256, 13560, -316160, 3830300, -25002240, 87841480, -180202240, 227671110, -180202240, 87841480, -25002240, 3830300, -316160, 13560, -256, 1},
{1, -1024, 255376, -30325760, 2060069240, -86239093760, 2306160223920, -40571580718080, 489632650203420, -4209374685189120, 26512089196724880, -124638699726597120, 442120325884773960, -1188638208146519040, 2420933452415430960, -3721572797083978752, 4298314898249481798, -3721572797083978752, 2420933452415430960, -1188638208146519040, 442120325884773960, -124638699726597120, 26512089196724880, -4209374685189120, 489632650203420, -40571580718080, 2306160223920, -86239093760, 2060069240, -30325760, 255376, -1024, 1}, ...

Crossrefs

Cf. A136674, A158800

Programs

  • Mathematica
    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] := {{2, -1}, {-3, 2}}
HadamardMatrix[n_] := Module[{m},
m = {{2, -1}, {-3, 2}};
KroneckerProduct[m, HadamardMatrix[n/2]]]
Table[HadamardMatrix[2^n], {n, 1, 4}]
Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[ HadamardMatrix[2^n], x], x], {n, 1, 6}]]
Flatten[%]

Formula

M(2)={{2, -1}, {-3, 2}};
M(4)={{4, -2, -2, 1}, {-6, 4, 3, -2}, {-6, 3, 4, -2}, {9, -6, -6, 4}},etc.

A173820 Coefficients of characteristic polynomials of Hadamard Cartan F_2 self-similar 2^n matrices:M={{2, -1}, {-2, 2}}.

Original entry on oeis.org

1, 2, -4, 1, 16, -64, 56, -16, 1, 4096, -32768, 75776, -77824, 39296, -9728, 1184, -64, 1, 4294967296, -68719476736, 375809638400, -1043677052928, 1696981843968, -1726845288448, 1143073669120, -506453819392, 152912134144, -31653363712
Offset: 0

Views

Author

Roger L. Bagula, Feb 25 2010

Keywords

Comments

Row sums are:
{1, -1, -7, -31, -208289151, 199276356275696712709633,
-27294457550222463310332530871924308277403810665846783,...}.

Examples

			{1},
{2, -4, 1},
{ 16, -64, 56, -16, 1},
{4096, -32768, 75776, -77824, 39296, -9728, 1184, -64, 1}, ...

Crossrefs

Cf. A136674, A158800, A173814, A136678

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] := {{2, -1}, {-2, 2}}
HadamardMatrix[n_] := Module[{m},
m = {{2, -1}, {-2, 2}};
KroneckerProduct[m, HadamardMatrix[n/2]]]
Table[HadamardMatrix[2^n], {n, 1, 4}]
Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[ HadamardMatrix[2^n], x], x], {n, 1, 6}]]
Flatten[%]

Formula

M(2)={{2, -1}, {-2, 2}};
M(4)={{4, -2, -2, 1}, {-4, 4, 2, -2}, {-4, 2, 4, -2}, {4, -4, -4, 4}},etc.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.