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.

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A137422 Triangle T(n,k) = A053120(n-1,k) + A053120(n-1,k-1), read by rows.

Original entry on oeis.org

0, 1, 1, 0, 1, 1, -1, -1, 2, 2, 0, -3, -3, 4, 4, 1, 1, -8, -8, 8, 8, 0, 5, 5, -20, -20, 16, 16, -1, -1, 18, 18, -48, -48, 32, 32, 0, -7, -7, 56, 56, -112, -112, 64, 64, 1, 1, -32, -32, 160, 160, -256, -256, 128, 128, 0, 9, 9, -120, -120, 432, 432, -576, -576, 256, 256
Offset: 0

Views

Author

Roger L. Bagula and Gary W. Adamson, Apr 16 2008

Keywords

Comments

Row sums are 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ...

Examples

			Triangle begins
   0;
   1,  1;
   0,  1,   1;
  -1, -1,   2,    2;
   0, -3,  -3,    4,    4;
   1,  1,  -8,   -8,    8,    8;
   0,  5,   5,  -20,  -20,   16,   16;
  -1, -1,  18,   18,  -48,  -48,   32,   32;
   0, -7,  -7,   56,   56, -112, -112,   64,   64;
   1,  1, -32,  -32,  160,  160, -256, -256,  128, 128;
   0,  9,   9, -120, -120,  432,  432, -576, -576, 256, 256;

Crossrefs

Cf. A053120.

Programs

  • Maple
    A053120 := proc(n, k)
    if n <0 or k <0 then
        0 ;
    else
        T(n, x) ;
        coeftayl(%, x=0, k) ;
    end if;
end proc:
A137422 := proc(n,k)
    A053120(n-1,k)+A053120(n-1,k-1)
end proc: # R. J. Mathar, Sep 10 2013
  • Mathematica
    (* Chebyshev A053120 polynomials*) (* Recursive root shifted polynomials*) Q[x, 0] = 1; Q[x, 1] = x + 1; Q[x_, n_] := (x + 1)*ChebyshevT[n - 1, x]; Table[ExpandAll[Q[x, n]], {n, 0, 10}]; a0 = Table[CoefficientList[Q[x, n], x], {n, 0, 10}]; Flatten[a0]

Extensions

T(0,0) set to a rationalized 0. - R. J. Mathar, Sep 10 2013

A137423 Triangle T(n,k) = A053120(n,k)+binomial(n,k) read by rows, 0<=k<=n.

Original entry on oeis.org

2, 1, 2, 0, 2, 3, 1, 0, 3, 5, 2, 4, -2, 4, 9, 1, 10, 10, -10, 5, 17, 0, 6, 33, 20, -33, 6, 33, 1, 0, 21, 91, 35, -91, 7, 65, 2, 8, -4, 56, 230, 56, -228, 8, 129, 1, 18, 36, -36, 126, 558, 84, -540, 9, 257, 0, 10, 95, 120, -190, 252, 1330, 120, -1235, 10, 513
Offset: 0

Views

Author

Roger L. Bagula and Gary W. Adamson, Apr 16 2008

Keywords

Examples

			2;
1, 2;
0, 2, 3;
1, 0, 3, 5;
2, 4, -2, 4, 9;
1, 10, 10, -10, 5, 17;
0, 6, 33, 20, -33, 6, 33;
1, 0, 21, 91, 35, -91, 7, 65;
2, 8, -4, 56, 230, 56, -228, 8, 129;
1, 18, 36, -36, 126, 558, 84, -540, 9, 257;
0, 10, 95, 120, -190, 252, 1330, 120, -1235, 10, 513;

Crossrefs

Cf. A053120, A000051 (row sums and diagonal)

Programs

  • Maple
    A137423 := proc(n,k)
    A053120(n,k)+binomial(n,k)
end proc: # R. J. Mathar, Sep 10 2013
  • Mathematica
    (* Chebyshev A053120 polynomials*) (* addition of coefficients of Polynomials*) Q[x, 0] = 2; Q[x, 1] = x + 1 + ChebyshevT[1, x]; Q[x_, n_] := (x + 1)^n + ChebyshevT[n, x]; Table[ExpandAll[Q[x, n]], {n, 0, 10}]; a0 = Table[CoefficientList[Q[x, n], x], {n, 0, 10}]; Flatten[a0]

A136160 Triangle T(n,k) = k*A053120(n,k).

Original entry on oeis.org

1, 0, 4, -3, 0, 12, 0, -16, 0, 32, 5, 0, -60, 0, 80, 0, 36, 0, -192, 0, 192, -7, 0, 168, 0, -560, 0, 448, 0, -64, 0, 640, 0, -1536, 0, 1024, 9, 0, -360, 0, 2160, 0, -4032, 0, 2304, 0, 100, 0, -1600, 0, 6720, 0, -10240, 0, 5120, -11, 0, 660, 0, -6160, 0, 19712, 0, -25344, 0, 11264
Offset: 1

Views

Author

Roger L. Bagula, Mar 16 2008

Keywords

Comments

The definition is equivalent to building the derivatives of the Chebyshev polynomials T(n,x) and listing the coefficients [x^k] dT/dx in row n.
Row sums are the squares A000079(n-1).
Obtained from A136265 by sign flips and nulling each second diagonal. - R. J. Mathar, Sep 04 2011

Examples

			1;
0, 4;
-3, 0, 12;
0, -16, 0, 32;
5, 0, -60, 0, 80;
0, 36, 0, -192, 0, 192;
-7, 0, 168, 0, -560, 0, 448;
0, -64, 0, 640, 0, -1536,0, 1024;
9, 0, -360, 0, 2160,0, -4032, 0, 2304;
0, 100, 0, -1600, 0, 6720, 0, -10240, 0, 5120;
-11, 0, 660, 0, -6160, 0, 19712, 0, -25344, 0, 11264;

References

  • Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986, page 8 and pages 42 - 43

Links

Crossrefs

Cf. A053120, A135929.

Programs

  • Mathematica
    P[x, 0] = 1; P[x, 1] = x; P[x_, n_] := P[x, n] = 2*x*P[x, n - 1] - P[x, n - 2]; Q[x_, n_] := D[P[x, n + 1], x]; a = Table[CoefficientList[Q[x, n], x], {n, 0, 10}]; Flatten[a]

A136163 Integration of A053120: triangle of coefficients of integration of Chebyshev's T(n,x) polynomials (powers of x in increasing order).

Original entry on oeis.org

1, -1, -1, -1, -3, 0, 2, 4, 0, -12, 0, 8, -1, 15, 0, -40, 0, 24, -4, 0, 60, 0, -120, 0, 64, -1, -35, 0, 210, 0, -336, 0, 160, 8, 0, -168, 0, 672, 0, -896, 0, 384, -1, 63, 0, -672, 0, 2016, 0, -2304, 0, 896, -8, 0, 360, 0, -2400, 0, 5760, 0, -5760, 0, 2048, -1, -99, 0, 1650, 0, -7920, 0, 15840, 0, -14080, 0, 4608
Offset: 1

Views

Author

Roger L. Bagula, Mar 16 2008, corrected Apr 06 2008

Keywords

Comments

The row sums are:
{-2, 0, -2, 0, -2, 0, -2, 0, -2, 0, -2}
These polynomials are orthogonal:
Table[Table[Integrate[Sqrt[1/(1 - x^2)]*a0[[ n]]*a0[[m]], {x, -1, 1}], {n, 1, 11}], {m, 1, 11}]
Solving for the recurrence:
Table[{c, d} /. Solve[{a0[[n]] -c*x*a0[[n - 1]] + d*a0[[n - 2]] == 0, a0[[n + 1]] - c*x*a0[[n]] + d*a0[[n - 1]] == 0}, {c, d}], {n, 3, 8}];
gives:
Q(x,n)=2*x*Q(x,n-1)-Q(x,n-2)

Examples

			{1},
{-1, -1},
{},
{-1, -3, 0, 2},
{4,0, -12, 0, 8},
{-1, 15, 0, -40, 0, 24},
{-4, 0, 60, 0, -120, 0, 64},
{-1, -35, 0, 210, 0, -336, 0, 160},
{8, 0, -168, 0,672, 0, -896, 0, 384},
{-1, 63, 0, -672, 0, 2016, 0, -2304, 0, 896}.
{-8, 0, 360, 0, -2400, 0, 5760, 0, -5760, 0, 2048},
{-1, -99, 0, 1650, 0, -7920, 0, 15840, 0, -14080, 0, 4608}

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795.
  • Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986, page 8 and pages 42 - 43;

Crossrefs

Cf. A053120.

Programs

  • Mathematica
    P[x, 0] = 1; P[x, 1] = x; P[x_, n_] := P[x, n] = 2*x*P[x, n - 1] - P[x, n - 2]; a0 = Table[ExpandAll[P[x, n]] /. x -> y, {n, 0, 10}]; b0 = Table[n*(n - 2)*Integrate[a0[[n]], {y, -1, x}], {n, 1, 11}] a = Join[{{1}}, Table[CoefficientList[b0[[n]], x], {n, 1, 11}]] Table[Apply[Plus, CoefficientList[b0[[n]], x]], {n, 1, 11}] Flatten[a]

Formula

T(x,n)=2*x*T(x,n-1)-T(x,n-2); Q(x,n)=Integrate[T(y,n-1),{y,-1,x}]

A136203 Derived Shabat linear tree transform of A053120: Triangle of coefficients of transformed Chebyshev's T(n, x) polynomials (powers of x in increasing order) T(x,n)->c*T(c*x+d)+d: c=-1;d=1; as substitution: 1-x->y( here alternative starting polynomial of Q(y,1]=1-y.

Original entry on oeis.org

1, 1, -1, 1, -2, 2, 1, -3, 8, -4, 1, -4, 20, -24, 8, 1, -5, 40, -84, 64, -16, 1, -6, 70, -224, 288, -160, 32, 1, -7, 112, -504, 960, -880, 384, -64, 1, -8, 168, -1008, 2640, -3520, 2496, -896, 128, 1, -9, 240, -1848, 6336, -11440, 11648, -6720, 2048, -256, 1, -10, 330, -3168, 13728, -32032, 43680, -35840, 17408
Offset: 1

Views

Author

Roger L. Bagula, Mar 16 2008

Keywords

Comments

The name contains unmatched parentheses. - Editors, Mar 13 2024
Row sums are:
{1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1}
integration is alternating on transformed {0,2} domain:
Table[Table[Integrate[Sqrt[1/(1 - (1 - y)^2)]*Q[y, n]*Q[y, m], {y, 0, 2}], {n, 0, 10}], {m, 0, 10}]

Examples

			{1},
{1, -1},
{1, -2, 2},
{1, -3, 8, -4},
{1, -4, 20, -24,8},
{1, -5, 40, -84, 64, -16},
{1, -6, 70, -224, 288, -160, 32},
{1, -7, 112, -504, 960, -880, 384, -64},
{1, -8, 168, -1008, 2640, -3520, 2496, -896, 128},
{1, -9, 240, -1848, 6336, -11440, 11648, -6720, 2048, -256},
{1, -10, 330, -3168, 13728, -32032, 43680, -35840, 17408, -4608, 512}

References

  • http://logic.pdmi.ras.ru/~yumat/personaljournal/chebyshev/chebysh.htm Quote: "... how many polynomials can generate given tree? It is easy to see that if P is a generalized Chebyshev Polynomial, then so is polynomial CP(cz+d)+D, moreover, it represents the same tree (of course, provided that both C and c are different from zero). In some natural sense these two linear transformations exhaust the variety of polynomials representing given tree. Namely, every drawing of a tree on the plane introduces an additional structure--circular order of edges around given vertex (say, clock-wise). Dealing with Chebyshev polynomials, it is natural to speak about plane trees understanding by them trees with this additional structure. "

Crossrefs

Cf. A053120.

Programs

  • Mathematica
    Clear[c, d, x0, x1, x2, P, Q, x, n, a] P[x, 0] = 1; P[x, 1] = x; P[x_, n_] := P[x, n] = 2*x*P[x, n - 1] - P[x, n - 2]; Solve[c*x0 + d - 2*x*(c*x1 + d) + c*x2 + d == 0, x0] c = -1; d = 1; (* Transform : c*x + d -> y*) Q[y, -1] = 0; Q[y, 0] = 1; Q[y, 1] =1- y; Q[y_, n_] := Q[y, n] = -(-2 + 2 (1 - y) - 2 (1 - y) Q[y, n - 1] + Q[y, n - 2]); Table[ExpandAll[Q[y, n]], {n, 0, 10}]; a = Table[CoefficientList[Q[y, n], y], {n, 0, 10}]; Flatten[a]

Formula

Q(y,0)=1;Q(y,1)=1-y; Q(y, n) = -(-2 + 2 (1 - y) - 2 (1 - y) Q(y, n - 1) + Q(y, n - 2))

A136665 Triangle of coefficients of Hermite-like analog of A053120 Chebyshev's T(n, x) polynomials (powers of x in increasing order): p(x,n)=2*x*p(x,n-1)-n*p(x,n-2).

Original entry on oeis.org

1, 0, 1, -2, 0, 2, 0, -7, 0, 4, 8, 0, -22, 0, 8, 0, 51, 0, -64, 0, 16, -48, 0, 234, 0, -176, 0, 32, 0, -453, 0, 916, 0, -464, 0, 64, 384, 0, -2778, 0, 3240, 0, -1184, 0, 128, 0, 4845, 0, -13800, 0, 10656, 0, -2944, 0, 256, -3840, 0, 37470, 0, -60000, 0, 33152, 0, -7168, 0, 512
Offset: 1

Views

Author

Roger L. Bagula, Apr 02 2008

Keywords

Comments

Row sums:
{1, 1, 0, -3, -6, 3, 42, 63, -210, -987, 126}

Examples

			{1},
{0, 1},
{-2, 0, 2},
{0, -7, 0, 4},
{8, 0, -22, 0, 8},
{0, 51, 0, -64, 0, 16},
{-48, 0, 234, 0, -176, 0, 32},
{0, -453, 0, 916, 0, -464,0, 64},
{384, 0, -2778, 0, 3240, 0, -1184, 0, 128},
{0, 4845, 0, -13800, 0, 10656, 0, -2944, 0,256},
{-3840, 0, 37470, 0, -60000, 0, 33152, 0, -7168, 0, 512}

Crossrefs

Cf. A053120.

Programs

  • Mathematica
    P[x, 0] = 1; P[x, 1] = x; P[x_, n_] := P[x, n] = 2*x*P[x, n - 1] - n*P[x, n - 2]; Table[ExpandAll[P[x, n]], {n, 0, 10}]; a = Table[CoefficientList[P[x, n], x], {n, 0, 10}]; Flatten[a]

Formula

p(x,n)=2*x*p(x,n-1)-n*p(x,n-2).

A137307 A triangular sequence of coefficients of even plus odd Chebyshev polynomials, A053120: q(x,n) = T(x,2*n-1)+T(x,2*n).

Original entry on oeis.org

1, 1, -1, 1, 2, 1, -3, -8, 4, 8, -1, 5, 18, -20, -48, 16, 32, 1, -7, -32, 56, 160, -112, -256, 64, 128, -1, 9, 50, -120, -400, 432, 1120, -576, -1280, 256, 512, 1, -11, -72, 220, 840, -1232, -3584, 2816, 6912, -2816, -6144, 1024, 2048, -1, 13, 98, -364, -1568, 2912, 9408, -9984, -26880, 16640, 39424, -13312, -28672, 4096, 8192
Offset: 1

Views

Author

Roger L. Bagula, Apr 20 2008

Keywords

Comments

The row sums are all 2 and double integrations are all orthogonal except for the zero to one level.
This arose from an idea of Chladni Chebyshev's:
q(exp(i*t),n) = T(cos(2*Pi*t),2*n-1)+T(sin(2*Pi*t),2*n)
which are strange looping spirals.

Examples

			Triangle begins:
  {1, 1},
  {-1, 1, 2},
  {1, -3, -8, 4, 8},
  {-1, 5, 18, -20, -48, 16, 32},
  {1, -7, -32, 56, 160, -112, -256, 64, 128},
  {-1, 9, 50, -120, -400, 432, 1120, -576, -1280, 256, 512},
  {1, -11, -72, 220, 840, -1232, -3584, 2816, 6912, -2816, -6144, 1024, 2048},
  {-1, 13, 98, -364, -1568, 2912, 9408, -9984, -26880, 16640, 39424, -13312, -28672, 4096, 8192},
  ...

Crossrefs

Cf. A053120.

Programs

  • Mathematica
    Q[x_, n_] := ChebyshevT[2*n - 1, x] + ChebyshevT[2*n, x]; Table[ExpandAll[Q[x, n]], {n, 0, 10}]; a0 = Table[CoefficientList[Q[x, n], x], {n, 0, 10}]; Flatten[a0]

Formula

q(x,n) = T(x,2*n-1)+T(x,2*n).

A137363 Triangular sequence of coefficients based on a Hilbert Transform of A053120: Chebyshev T(x,n); Coefficients(A053120[n,m])-Floor[Imaginary part of( HilbertTransform(A053120(n,m))];.

Original entry on oeis.org

1, 0, 1, -1, -1, 2, 4, -3, -3, 4, 1, 6, -8, -9, 8, 7, 5, 15, -20, -20, 16, -1, -3, 18, 37, -48, -46, 32, 26, -6, -19, 57, 95, -112, -99, 64, 1, 16, -32, -80, 160, 233, -256, -213, 128, 86, 9, 54, -120, -254, 432, 566, -576, -450, 256, -1, 14, 50, 174, -400, -746, 1120, 1344, -1280, -947, 512
Offset: 1

Views

Author

Roger L. Bagula, Apr 26 2008

Keywords

Comments

Row sums are:
{1, 1, 0, 2, -2, 3, -11, 6, -43, 3, -160}
This Hilbert transform/ operator has the property
that to sign the a-b and a+b and the absolute value row sum for both is: ( called isobaric by Olver)
{1, 1, 4, 14, 32, 83, 185, 478, 1119, 2803, 6588}

Examples

			a-b:
{1},
{0, 1},
{-1, -1, 2},
{4, -3, -3, 4},
{1, 6, -8, -9, 8},
{7, 5, 15, -20, -20, 16},
{-1, -3, 18, 37, -48, -46, 32},
{26, -6, -19, 57, 95, -112, -99, 64},
{1, 16, -32, -80,160, 233, -256, -213, 128},
{86, 9, 54, -120, -254, 432, 566, -576, -450,256},
{-1, 14, 50, 174, -400, -746, 1120, 1344, -1280, -947, 512}
a+b:
{1},
{0, 1},
{-1, 1, 2},
{-4, -3, 3, 4},
{1, -6, -8, 9, 8},
{-7, 5, -15, -20, 20, 16},
{-1, 3, 18, -37, -48, 46, 32},
{-26, -8, 19, 55, -95, -112,99, 64},
{1, -16, -32, 80, 160, -233, -256, 213, 128},
{-86, 9, -54, -120, 254, 432, -566, -576, 450, 256},
{-1, -14, 50, -174, -400,746, 1120, -1344, -1280, 947, 512}

References

  • Wilbur R. LePage, Complex Variables and the Laplace Transform for Engineers,Dover, New York,1961, page 225.
  • P. J. Olver, Classical Invariant Theory, Cambridge Univ. Press, p. 222.
  • http://jowett.home.cern.ch/jowett/Mathematica/Accelerator/Hilbert.nb

Crossrefs

Cf. A053120.

Programs

  • Mathematica
    HilbertTransform[x_List] := Module[{nx, n, y}, nx = Length[x]; xn = If[EvenQ[nx], x, Append[x, 0]]; n = Length[xn]; y = Fourier[xn]; h = Flatten[{1, Table[2, {k, 2, n/2}], 1, Table[0, {k, n/2 + 2, n}]}]; Take[InverseFourier[h y], nx]]; a = Table[CoefficientList[ChebyshevT[n, x], x], {n, 0, 10}]; b = Table[Floor[Im[ HilbertTransform[CoefficientList[ChebyshevT[n, x], x]]]], {n, 0, 10}]; a-b

Formula

Coefficients(A053120[n,m])-Floor[Imaginary part of( HilbertTransform(A053120(n,m))];

A137430 Triangular sequence from coefficients of a cumulative sum of Chebyshev T(x,n) polynomials (A053120): p(x,n)=p(x,n-1)+T(x,n).

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 0, -2, 2, 4, 1, -2, -6, 4, 8, 1, 3, -6, -16, 8, 16, 0, 3, 12, -16, -40, 16, 32, 0, -4, 12, 40, -40, -96, 32, 64, 1, -4, -20, 40, 120, -96, -224, 64, 128, 1, 5, -20, -80, 120, 336, -224, -512, 128, 256, 0, 5, 30, -80, -280, 336, 896, -512, -1152, 256, 512
Offset: 1

Views

Author

Roger L. Bagula, Apr 27 2008

Keywords

Comments

Row sums are: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}

Examples

			{1},
{1, 1},
{0, 1, 2},
{0, -2, 2, 4},
{1, -2, -6, 4, 8},
{1, 3, -6, -16, 8, 6},
{0, 3, 12, -16, -40, 16, 32},
{0, -4, 12, 40, -40, -96, 32, 64},
{1, -4, -20, 40, 120, -96, -224, 64, 128},
{1, 5, -20, -80, 120, 336, -224, -512, 128, 256},
{0, 5, 30, -80, -280, 336,896, -512, -1152, 256, 512}

Crossrefs

Cf. A053120.

Programs

  • Mathematica
    Clear[P] P[x, -1] = 0; P[x, 0] = 1; P[x_, n_] := P[x, n] = P[x, n - 1] + ChebyshevT[n, x]; Table[P[x, n], {n, 0, 10}]; a = Table[CoefficientList[P[x, n], x], {n, 0, 10}]; Flatten[a]

Formula

p(x,n)=p(x,n-1)+T(x,n); out_n,m=Coefficients(p(x,n)).

A137448 Triangle T(n,k) = (1-k*(k-1))*A053120(n,k), read by rows, 0<=k<=n.

Original entry on oeis.org

1, 0, 1, -1, 0, -2, 0, -3, 0, -20, 1, 0, 8, 0, -88, 0, 5, 0, 100, 0, -304, -1, 0, -18, 0, 528, 0, -928, 0, -7, 0, -280, 0, 2128, 0, -2624, 1, 0, 32, 0, -1760, 0, 7424, 0, -7040, 0, 9, 0, 600, 0, -8208, 0, 23616, 0, -18176, -1, 0, -50, 0, 4400, 0, -32480, 0, 70400, 0, -45568
Offset: 0

Views

Author

Roger L. Bagula and Gary W. Adamson, Apr 18 2008

Keywords

Comments

The entries may also be defined as the coefficient [x^k] of the polynomial T(n,x)-x^2* (d^2/dx^2) T(n,x) where T are the Chebyshev polynomials (A053120).
Row sums are 1, 1, -3, -23, -79, -199, -419, -783, -1343, -2159, -3299, ...

Examples

			1;
0, 1;
-1, 0, -2;
0, -3, 0, -20;
1, 0, 8, 0, -88;
0, 5, 0, 100, 0, -304;
-1, 0, -18, 0, 528, 0, -928;
0, -7, 0, -280, 0, 2128, 0, -2624;
1, 0, 32, 0, -1760, 0, 7424, 0, -7040;
0, 9, 0, 600, 0, -8208,0, 23616, 0, -18176;
-1, 0, -50, 0, 4400, 0, -32480, 0, 70400, 0, -45568;

Crossrefs

Cf. A053120, A112742.

Programs

  • Mathematica
    Clear[p, x, a] p[x, 0] = 1; p[x, 1] = x + 1; p[x_, n_] := p[x, n] = -x^2*D[ChebyshevT[n, x], {x, 2}] + ChebyshevT[n, x]; Table[Expand[p[x, n]], {n, 0, 10}]; a = Table[CoefficientList[p[x, n], x], {n, 0, 10}]; Flatten[a]

Formula

Row sums: Sum_{k=0..n} T(n,k) = (3+n^2-n^4)/3 = 1-A112742(n). - R. J. Mathar, Sep 10 2013

Extensions

T(1,0) corrected by R. J. Mathar, Sep 10 2013
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