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 1-5 of 5 results.

A215064 Triangle read by rows, e.g.f. exp(x*z)*((exp(x/2)+exp(x*3/2))/((exp(3*x/2)+ 2*cos(sqrt(3)*x/2))/3)-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, -1, 3, 3, 1, -3, -4, 6, 4, 1, -9, -15, -10, 10, 5, 1, 19, -54, -45, -20, 15, 6, 1, 99, 133, -189, -105, -35, 21, 7, 1, 477, 792, 532, -504, -210, -56, 28, 8, 1, -1513, 4293, 3564, 1596, -1134, -378, -84, 36, 9, 1, -11259
Offset: 0

Views

Author

Peter Luschny, Aug 01 2012

Keywords

Examples

			[0] [1]
[1] [1, 1]
[2] [1, 2, 1]
[3] [-1, 3, 3, 1]
[4] [-3, -4, 6, 4, 1]
[5] [-9, -15, -10, 10, 5, 1]
[6] [19, -54, -45, -20, 15, 6, 1]
[7] [99, 133, -189, -105, -35, 21, 7, 1]
[8] [477, 792, 532, -504, -210, -56, 28, 8, 1]
[9] [-1513, 4293, 3564, 1596, -1134, -378, -84, 36, 9, 1]

Crossrefs

Cf. A215060, A215061, A215062, A215063, A215065.

Programs

  • Mathematica
    max = 11; f = Exp[x*z]*((Exp[x/2] + Exp[x*(3/2)])/((Exp[3*(x/2)] + 2*Cos[Sqrt[3]*(x/2)])/3) - 1); coes = CoefficientList[ Series[f, {x, 0, max}, {z, 0, max}], {x, z}]; Table[ coes[[n, k]]*(n - 1)!, {n, 1, max}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 29 2013 *)
  • Sage
    # uses[triangle from A215060]
def A215064_triangle(dim):
    var('x, z')
    f = exp(x*z)*((exp(x/2)+exp(x*3/2))/((exp(3*x/2)+2*cos(sqrt(3)*x/2))/3)-1)
    return triangle(f, dim)
A215064_triangle(12)

Formula

Matrix inverse is A215065.
T(n,k) = A215060(n,k) + A215062(n,k) - [n==k].
|T(n,0)| = A178963(n).
|T(3*n,0)| = A002115(n).

A215060 Triangle read by rows, e.g.f. exp(x*(z+1/2))/((exp(3*x/2) + 2*cos(sqrt(3)*x/2))/3).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, -1, 0, 0, 1, 0, -4, 0, 0, 1, 0, 0, -10, 0, 0, 1, 19, 0, 0, -20, 0, 0, 1, 0, 133, 0, 0, -35, 0, 0, 1, 0, 0, 532, 0, 0, -56, 0, 0, 1, -1513, 0, 0, 1596, 0, 0, -84, 0, 0, 1, 0, -15130, 0, 0, 3990, 0, 0, -120, 0, 0, 1, 0, 0, -83215, 0
Offset: 0

Views

Author

Peter Luschny, Aug 01 2012

Keywords

Examples

			[0] [1]
[1] [0, 1]
[2] [0, 0, 1]
[3] [-1, 0, 0, 1]
[4] [0, -4, 0, 0, 1]
[5] [0, 0, -10, 0, 0, 1]
[6] [19, 0, 0, -20, 0, 0, 1]
[7] [0, 133, 0, 0, -35, 0, 0, 1]
[8] [0, 0, 532, 0, 0, -56, 0, 0, 1]
[9] [-1513, 0, 0, 1596, 0, 0, -84, 0, 0, 1]

Crossrefs

Cf. A215061, A215062, A215063, A215064, A215065.

Programs

  • Sage
    def triangle(f, dim):
    var('x,z')
    s = f.series(x, dim+2)
    P = [factorial(i)*s.coefficient(x,i) for i in range(dim)]
    for k in range(dim): print([k], [P[k].coefficient(z,i) for i in (0..k)])
def A215060_triangle(dim) :
    var('x, z')
    f = exp(x*(z+1/2))/((exp(3*x/2)+2*cos(sqrt(3)*x/2))/3)
    return triangle(f, dim)
A215060_triangle(12)

Formula

Matrix inverse is A215061.
T(n,k) = A215064(n,k) - A215062(n,k) + [n==k].
|T(3*n,0)| = A002115(n).

A215062 Triangle read by rows, e.g.f. exp(x*(z+3/2))/((exp(3*x/2)+2*cos(sqrt(3)*x/2))/3).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 0, 3, 3, 1, -3, 0, 6, 4, 1, -9, -15, 0, 10, 5, 1, 0, -54, -45, 0, 15, 6, 1, 99, 0, -189, -105, 0, 21, 7, 1, 477, 792, 0, -504, -210, 0, 28, 8, 1, 0, 4293, 3564, 0, -1134, -378, 0, 36, 9, 1, -11259, 0, 21465, 11880, 0, -2268
Offset: 0

Views

Author

Peter Luschny, Aug 01 2012

Keywords

Examples

			[0] [1]
[1] [1, 1]
[2] [1, 2, 1]
[3] [0, 3, 3, 1]
[4] [-3, 0, 6, 4, 1]
[5] [-9, -15, 0, 10, 5, 1]
[6] [0, -54, -45, 0, 15, 6, 1]
[7] [99, 0, -189, -105, 0, 21, 7, 1]
[8] [477, 792, 0, -504, -210, 0, 28, 8, 1]
[9] [0, 4293, 3564, 0, -1134, -378, 0, 36, 9, 1]

Crossrefs

Cf. A215060, A215061, A215063, A215064, A215065.

Programs

  • Mathematica
    max = 11; f = Exp[x*(z + 3/2)]/((Exp[3*(x/2)] + 2*Cos[Sqrt[3]*(x/2)])/3); coes = CoefficientList[ Series[f, {x, 0, max}, {z, 0, max}], {x, z}]; Table[ coes[[n, k]]*(n-1)!, {n, 1, max}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 29 2013 *)
  • Sage
    def A215062_triangle(dim): # See A215060 for function 'triangle'.
    var('x, z')
    f = exp(x*(z+3/2))/((exp(3*x/2)+2*cos(sqrt(3)*x/2))/3)
    return triangle(f, dim)
A215062_triangle(12)

Formula

Matrix inverse is A215063.
T(n,k) = A215064(n,k) - A215060(n,k) + [n==k]

A215063 Triangle read by rows, e.g.f. exp(x*(z-3/2))*(exp(3*x/2)+2*cos(sqrt(3)*x/2))/3.

Original entry on oeis.org

1, -1, 1, 1, -2, 1, 0, 3, -3, 1, -3, 0, 6, -4, 1, 9, -15, 0, 10, -5, 1, -18, 54, -45, 0, 15, -6, 1, 27, -126, 189, -105, 0, 21, -7, 1, -27, 216, -504, 504, -210, 0, 28, -8, 1, 0, -243, 972, -1512, 1134, -378, 0, 36, -9, 1, 81, 0, -1215
Offset: 0

Views

Author

Peter Luschny, Aug 01 2012

Keywords

Comments

Matrix inverse is A215062.

Examples

			[0] [1]
[1] [-1, 1]
[2] [1, -2, 1]
[3] [0, 3, -3, 1]
[4] [-3, 0, 6, -4, 1]
[5] [9, -15, 0, 10, -5, 1]
[6] [-18, 54, -45, 0, 15, -6, 1]
[7] [27, -126, 189, -105, 0, 21, -7, 1]
[8] [-27, 216, -504, 504, -210, 0, 28, -8, 1]
[9] [0, -243, 972, -1512, 1134, -378, 0, 36, -9, 1]

Crossrefs

Cf. A215060, A215061, A215062, A215064, A215065.

Programs

  • Sage
    def A215063_triangle(dim): # See A215060 for function 'triangle'.
    var('x, z')
    f = exp(x*(z-3/2))*((exp(3*x/2)+2*cos(sqrt(3)*x/2))/3)
    return triangle(f, dim)
A215063_triangle(12)

A215065 Triangle read by rows, e.g.f. exp(x*z)/((exp(x/2)+exp(x*3/2))/((exp(3*x/2)+2*cos(sqrt(3)*x/2))/3)-1).

Original entry on oeis.org

1, -1, 1, 1, -2, 1, 1, 3, -3, 1, -11, 4, 6, -4, 1, 49, -55, 10, 10, -5, 1, -137, 294, -165, 20, 15, -6, 1, -127, -959, 1029, -385, 35, 21, -7, 1, 5573, -1016, -3836, 2744, -770, 56, 28, -8, 1, -50399, 50157, -4572, -11508, 6174, -1386, 84
Offset: 0

Views

Author

Peter Luschny, Aug 01 2012

Keywords

Comments

Matrix inverse is A215064.

Examples

			[0] [1]
[1] [-1, 1]
[2] [1, -2, 1]
[3] [1, 3, -3, 1]
[4] [-11, 4, 6, -4, 1]
[5] [49, -55, 10, 10, -5, 1]
[6] [-137, 294, -165, 20, 15, -6, 1]
[7] [-127, -959, 1029, -385, 35, 21, -7, 1]
[8] [5573, -1016, -3836, 2744, -770, 56, 28, -8, 1]
[9] [-50399, 50157, -4572, -11508, 6174, -1386, 84, 36, -9, 1]

Crossrefs

Cf. A215060, A215061, A215062, A215063, A215064.

Programs

  • Mathematica
    max = 10; f = Exp[x*z]/((Exp[x/2] + Exp[x*(3/2)])/((Exp[3*(x/2)] + 2*Cos[Sqrt[3]*(x/2)])/3) - 1); coes = CoefficientList[ Series[f, {x, 0, max}, {z, 0, max}], {x, z}]; Table[ coes[[n, k]]*(n - 1)!, {n, 1, max}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 29 2013 *)
  • Sage
    def A215065_triangle(dim): # See A215060 for function 'triangle'.
    var('x, z')
    f = exp(x*z)/((exp(x/2)+exp(x*3/2))/((exp(3*x/2)+2*cos(sqrt(3)*x/2))/3)-1)
    return triangle(f, dim)
A215065_triangle(12)
Showing 1-5 of 5 results.

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

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