A110162 -id:A110162 - cp's OEIS frontend

A128411 Coefficient array for orthogonal polynomials defined by C(2n,n).

1, -2, 1, 4, -8, 2, -8, 36, -24, 4, 16, -128, 160, -64, 8, -32, 400, -800, 560, -160, 16, 64, -1152, 3360, -3584, 1728, -384, 32, -128, 3136, -12544, 18816, -13440, 4928, -896, 64, 256, -8192, 43008, -86016, 84480, -45056, 13312

Offset: 0

Paul Barry, Mar 02 2007

Define {p(n,x)} to be the family of orthogonal polynomials on [0,4] for the weight function (1/pi)*1/sqrt(x(4-x)) which defines C(2n,n). We have p(n,x)=(2x-4)*p(n-1,x)-4*p(n-2,x), with p(0,x)=1, p(1,x)=-2+x. A scaled version of this triangle is given by A128412.

			Triangle begins
1,
-2, 1,
4, -8, 2,
-8, 36, -24, 4,
16, -128, 160, -64, 8,
-32, 400, -800, 560, -160, 16,
64, -1152, 3360, -3584, 1728, -384, 32

Column k has g.f. if(k=0,1/(1+2x),(1-2x)*((2^(k-1)+0^k/2)*x^k/(1+2x)^(2k+1))).

T(n,k)=(C(n+k,n-k)(-1)^(n-k)-C(n+k-1,n-k-1)(-1)^(n-k-1))*(2^(n-1)+0^n/2); T(n,k)=A110162(n,k)*(2^(n-1)+0^n/2); - Paul Barry, Mar 22 2007

A136321 Triangular sequence of coefficients of a polynomial recursion for C_n and B_n Cartan matrices: p(x, n) = (-2 + x)p(x, n - 1) - p(x, n - 2) p(x,n)=x2-4x+4-m:m=5;(related sequence: A_n:m=1,G_n,m=3,B_n,C_n,m=2) This triangular sequence is an extension to the Cartan pattern of matrices.

Original entry on oeis.org

1, -2, 1, -1, -4, 1, 4, 6, -6, 1, -7, -4, 17, -8, 1, 10, -5, -32, 32, -10, 1, -13, 24, 42, -88, 51, -12, 1, 16, -56, -28, 186, -180, 74, -14, 1, -19, 104, -42, -312, 495, -316, 101, -16, 1, 22, -171, 216, 396, -1122, 1053, -504, 132, -18, 1, -25, 260, -561, -264, 2145, -2912, 1960, -752, 167, -20, 1

Offset: 1

Roger L. Bagula, Apr 12 2008

Row sums are:
{1, -1, -4, 5, -1, -4, 5, -1, -4, 5, -1}
This sequence is also related to different p(x,2) start:
1) A_n like sequence A053122 ( sign change)
2) my G_n matrix A136674
3) B_n,C_n A110162

			{1},
{-2, 1},
{-1, -4, 1},
{4, 6, -6, 1},
{-7, -4, 17, -8, 1},
{10, -5, -32, 32, -10, 1},
{-13, 24, 42, -88,51, -12, 1},
{16, -56, -28,186, -180, 74, -14, 1},
{-19, 104, -42, -312, 495, -316, 101, -16, 1},
{22, -171, 216, 396, -1122, 1053, -504, 132, -18, 1},
{-25, 260, -561, -264,2145, -2912, 1960, -752, 167, -20, 1}

Cf. A053122, A136674, A110162.

Clear[p, a] p[x, 0] = 1; p[x, 1] = -2 + x; p[x, 2] = x^2 - 4*x - 1; p[x_, n_] := p[x, n] = (-2 + x)*p[x, n - 1] - p[x, n - 2]; Table[ExpandAll[p[x, n]], {n, 0, 10}]; a = Table[CoefficientList[p[x, n], x], {n, 0, 10}] Flatten[a]

p(x, n) = (-2 + x)*p(x, n - 1) - p(x, n - 2) Three start vectors necessary: p(x,0)=1;p(x,1)=2-x; p(x,2)=x^2-4*x-1=CharacteristicPolynomial[{{2, -5}, {-1, 2}}, x] or CharacteristicPolynomial[{{2, -1}, {-5, 2}}, x]

A136329 Triangular sequence of coefficients of a polynomial recursion for C_n and B_n Cartan matrices: p(x, n) = (-2 + x)p(x, n - 1) - p(x, n - 2) p(x,n)=x2-4x+4-m:m=4;(related sequence: A_n:m=1,G_n,m=3,B_n,C_n,m=2) This triangular sequence is an extension to the Cartan pattern of matrices.

Original entry on oeis.org

1, -2, 1, 0, -4, 1, 2, 7, -6, 1, -4, -8, 18, -8, 1, 6, 5, -38, 33, -10, 1, -8, 4, 63, -96, 52, -12, 1, 10, -21, -84, 222, -190, 75, -14, 1, -12, 48, 84, -432, 550, -328, 102, -16, 1, 14, -87, -36, 726, -1342, 1131, -518, 133, -18, 1, -16, 140, -99, -1056, 2860, -3276, 2065, -768, 168, -20, 1

Offset: 1

Roger L. Bagula, Apr 12 2008

Row sums are:
{1, -1, -3, 4, -1, -3, 4, -1, -3, 4, -1}
This sequence is also related to different p(x,2) start:
1) A_n like sequence A053122 ( sign change)
2) my G_n matrix A136674
3) B_n,C_n A110162

			{1},
{-2, 1},
{0, -4, 1},
{2, 7, -6, 1},
{-4, -8, 18, -8, 1},
{6, 5, -38, 33, -10,1},
{-8, 4, 63, -96, 52, -12, 1},
{10, -21, -84, 222, -190, 75, -14, 1},
{-12, 48, 84, -432, 550, -328, 102, -16, 1},
{14, -87, -36, 726, -1342, 1131, -518, 133, -18, 1},
{-16, 140, -99, -1056, 2860, -3276, 2065, -768, 168, -20, 1}

Cf. A053122, A136674, A110162.

Clear[p, a] p[x, 0] = 1; p[x, 1] = -2 + x; p[x, 2] = x^2 - 4*x ; p[x_, n_] := p[x, n] = (-2 + x)*p[x, n - 1] - p[x, n - 2]; Table[ExpandAll[p[x, n]], {n, 0, 10}]; a = Table[CoefficientList[p[x, n], x], {n, 0, 10}] Flatten[a]

p(x, n) = (-2 + x)*p(x, n - 1) - p(x, n - 2) Three start vectors necessary: p(x,0)=1;p(x,1)=2-x; p(x,2)=x^2-4*x=CharacteristicPolynomial[{{2, -4}, {-1, 2}}, x] or CharacteristicPolynomial[{{2, -1}, {-4, 2}}, x]

cp's OEIS Frontend

A128411 Coefficient array for orthogonal polynomials defined by C(2n,n).

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A136321 Triangular sequence of coefficients of a polynomial recursion for C_n and B_n Cartan matrices: p(x, n) = (-2 + x)p(x, n - 1) - p(x, n - 2) p(x,n)=x2-4x+4-m:m=5;(related sequence: A_n:m=1,G_n,m=3,B_n,C_n,m=2) This triangular sequence is an extension to the Cartan pattern of matrices.

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Formula

A136329 Triangular sequence of coefficients of a polynomial recursion for C_n and B_n Cartan matrices: p(x, n) = (-2 + x)p(x, n - 1) - p(x, n - 2) p(x,n)=x2-4x+4-m:m=4;(related sequence: A_n:m=1,G_n,m=3,B_n,C_n,m=2) This triangular sequence is an extension to the Cartan pattern of matrices.

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Mathematica

Formula

cp's OEIS Frontend

A128411 Coefficient array for orthogonal polynomials defined by C(2n,n).

Views

Author

Keywords

Comments

Examples

Formula

A136321 Triangular sequence of coefficients of a polynomial recursion for C_n and B_n Cartan matrices: p(x, n) = (-2 + x)*p(x, n - 1) - p(x, n - 2) p(x,n)=x2-4*x+4-m:m=5;(related sequence: A_n:m=1,G_n,m=3,B_n,C_n,m=2) This triangular sequence is an extension to the Cartan pattern of matrices.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A136329 Triangular sequence of coefficients of a polynomial recursion for C_n and B_n Cartan matrices: p(x, n) = (-2 + x)*p(x, n - 1) - p(x, n - 2) p(x,n)=x2-4*x+4-m:m=4;(related sequence: A_n:m=1,G_n,m=3,B_n,C_n,m=2) This triangular sequence is an extension to the Cartan pattern of matrices.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A136321 Triangular sequence of coefficients of a polynomial recursion for C_n and B_n Cartan matrices: p(x, n) = (-2 + x)p(x, n - 1) - p(x, n - 2) p(x,n)=x2-4x+4-m:m=5;(related sequence: A_n:m=1,G_n,m=3,B_n,C_n,m=2) This triangular sequence is an extension to the Cartan pattern of matrices.

A136329 Triangular sequence of coefficients of a polynomial recursion for C_n and B_n Cartan matrices: p(x, n) = (-2 + x)p(x, n - 1) - p(x, n - 2) p(x,n)=x2-4x+4-m:m=4;(related sequence: A_n:m=1,G_n,m=3,B_n,C_n,m=2) This triangular sequence is an extension to the Cartan pattern of matrices.