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 1-7 of 7 results.

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).

A215061 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, 1, 0, 0, 20, 0, 0, 1, 0, 7, 0, 0, 35, 0, 0, 1, 0, 0, 28, 0, 0, 56, 0, 0, 1, 1, 0, 0, 84, 0, 0, 84, 0, 0, 1, 0, 10, 0, 0, 210, 0, 0, 120, 0, 0, 1, 0, 0, 55, 0, 0, 462, 0, 0, 165, 0, 0, 1, 1, 0, 0
Offset: 0

Views

Author

Peter Luschny, Aug 01 2012

Keywords

Comments

Matrix inverse is A215060.

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] [1, 0, 0, 20, 0, 0, 1]
[7] [0, 7, 0, 0, 35, 0, 0, 1]
[8] [0, 0, 28, 0, 0, 56, 0, 0, 1]
[9] [1, 0, 0, 84, 0, 0, 84, 0, 0, 1]

Crossrefs

Cf. A215060, A215062, A215063, A215064, A215065.

Programs

  • Sage
    def A215061_triangle(dim): # See A215060 for function 'triangle'.
    var('x, z')
    f = exp(x*(z-1/2))*((exp(3*x/2)+2*cos(sqrt(3)*x/2))/3)
    return triangle(f, dim)
A215061_triangle(12)

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)

A281587 Triangle read by rows, e.g.f. exp(x*z)*(2*(exp(z)+1)/(cosh(z)+cos(z))-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, -1, 4, 6, 4, 1, -4, -5, 10, 10, 5, 1, -14, -24, -15, 20, 15, 6, 1, -34, -98, -84, -35, 35, 21, 7, 1, 69, -272, -392, -224, -70, 56, 28, 8, 1, 496, 621, -1224, -1176, -504, -126, 84, 36, 9, 1, 2896, 4960, 3105, -4080, -2940, -1008, -210, 120, 45, 10, 1
Offset: 0

Views

Author

Peter Luschny, Feb 01 2017

Keywords

Examples

			Triangle starts:
[   1]
[   1,   1]
[   1,   2,   1]
[   1,   3,   3,   1]
[  -1,   4,   6,   4,  1]
[  -4,  -5,  10,  10,  5,  1]
[ -14, -24, -15,  20, 15,  6, 1]
[ -34, -98, -84, -35, 35, 21, 7, 1]

Crossrefs

Cf. A130595 (m=1), A109449 (m=2), A215064 (m=3), A281587 (m=4).
Cf. A178964 (col. 0), A281588 (col. 1).

Programs

  • Maple
    A281587_row := proc(n) exp(x*z)*(2*(exp(z)+1)/(cosh(z)+cos(z))-1);
n!*coeff(series(%,z,n+1),z,n); PolynomialTools:-CoefficientList(%,x) end:
for n from 0 to 12 do A281587_row(n) od;

A281586 E.g.f.: z*(((exp(z)+1)/((exp(z)+2*exp(-z/2)*cos(z*sqrt(3)/2))/3))-1).

Original entry on oeis.org

0, 1, 2, 3, -4, -15, -54, 133, 792, 4293, -15130, -123849, -895212, 4101799, 42732522, 386103915, -2177360656, -27544520691, -298649804706, 1999963458217, 29765143177020, 376514568163377, -2919514870785766, -49968285360277437, -722370834074695896, 6365117686550339275
Offset: 0

Views

Author

Peter Luschny, Jan 25 2017

Keywords

Crossrefs

a(n) = A215064(n,1).

Programs

  • Maple
    A281586_list := proc(len) z*(((exp(z)+1)/((exp(z)+2*exp(-z/2)* cos(z*sqrt(3)/2))/3))-1); series(%,z,len+1); seq(k!*coeff(%,z,k),k=0..len) end: A281586_list(20);
  • Sage
    # uses[SIB from A281588]
A281586_list = lambda len: SIB(len, 3)
print(A281586_list(26))
Showing 1-7 of 7 results.

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

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