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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A265604 Triangle read by rows: The inverse Bell transform of the quartic factorial numbers (A007696).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, -2, 3, 1, 0, 10, -5, 6, 1, 0, -80, 30, -5, 10, 1, 0, 880, -290, 45, 5, 15, 1, 0, -12320, 3780, -560, 35, 35, 21, 1, 0, 209440, -61460, 8820, -735, 0, 98, 28, 1, 0, -4188800, 1192800, -167300, 14700, -735, 0, 210, 36, 1
Offset: 0

Views

Author

Peter Luschny, Dec 30 2015

Keywords

Examples

			[ 1]
[ 0,      1]
[ 0,      1,      1]
[ 0,     -2,      3,      1]
[ 0,     10,     -5,      6,      1]
[ 0,    -80,     30,     -5,     10,      1]
[ 0,    880,   -290,     45,      5,     15,      1]

Links

Crossrefs

Cf. A007696, A264428, A264429.
Inverse Bell transforms of other multifactorials are: A048993, A049404, A049410, A075497, A075499, A075498, A119275, A122848, A265605.

Programs

  • Sage
    # uses[bell_transform from A264428]
def inverse_bell_matrix(generator, dim):
    G = [generator(k) for k in srange(dim)]
    row = lambda n: bell_transform(n, G)
    M = matrix(ZZ, [row(n)+[0]*(dim-n-1) for n in srange(dim)]).inverse()
    return matrix(ZZ, dim, lambda n,k: (-1)^(n-k)*M[n,k])
multifact_4_1 = lambda n: prod(4*k + 1 for k in (0..n-1))
print(inverse_bell_matrix(multifact_4_1, 8))

A265605 Triangle read by rows: The inverse Bell transform of the triple factorial numbers (A007559).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, -1, 3, 1, 0, 3, -1, 6, 1, 0, -15, 5, 5, 10, 1, 0, 105, -35, 0, 25, 15, 1, 0, -945, 315, -35, 0, 70, 21, 1, 0, 10395, -3465, 490, -35, 70, 154, 28, 1, 0, -135135, 45045, -6895, 630, -105, 378, 294, 36, 1
Offset: 0

Views

Author

Peter Luschny, Dec 30 2015

Keywords

Examples

			[ 1]
[ 0,    1]
[ 0,    1,    1]
[ 0,   -1,    3,    1]
[ 0,    3,   -1,    6,    1]
[ 0,  -15,    5,    5,   10,    1]
[ 0,  105,  -35,    0,   25,   15,    1]
[ 0, -945,  315,  -35,    0,   70,   21,    1]

Links

Crossrefs

Cf. A007559, A264428, A264429.
Inverse Bell transforms of other multifactorials are: A048993, A049404, A049410, A075497, A075499, A075498, A119275, A122848, A265604.

Programs

  • Sage
    # uses[bell_transform from A264428]
def inverse_bell_matrix(generator, dim):
    G = [generator(k) for k in srange(dim)]
    row = lambda n: bell_transform(n, G)
    M = matrix(ZZ, [row(n)+[0]*(dim-n-1) for n in srange(dim)]).inverse()
    return matrix(ZZ, dim, lambda n,k: (-1)^(n-k)*M[n,k])
multifact_3_1 = lambda n: prod(3*k + 1 for k in (0..n-1))
print(inverse_bell_matrix(multifact_3_1, 8))

A019677 Expansion of 1/((1-4x)(1-8x)(1-12x)).

Original entry on oeis.org

1, 24, 400, 5760, 77056, 989184, 12390400, 152862720, 1867841536, 22682271744, 274333696000, 3309180026880, 39847582498816, 479270434504704, 5760041038643200, 69190860134154240, 830853267268304896, 9974742789667160064, 119732942204305408000
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Third column of triangle A075499.
Cf. A016152, A075907.

Programs

  • Magma
    m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-4*x)*(1-8*x)*(1-12*x)))); /* or */ I:=[1, 24, 400]; [n le 3 select I[n] else 24*Self(n-1)-176*Self(n-2)+384*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Jul 03 2013
  • Maple
    a:= n-> (Matrix(3, (i, j)-> `if`(i=j-1, 1, `if`(j=1, [24, -176, 384][i], 0)))^n)[1, 1]: seq(a(n), n=0..25);  # Alois P. Heinz, Jul 03 2013
  • Mathematica
    CoefficientList[Series[1 / ((1 - 4 x) (1 - 8 x) (1 - 12 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Jul 03 2013 *)
LinearRecurrence[{24,-176,384},{1,24,400},20] (* Harvey P. Dale, Jul 18 2020 *)
  • PARI
    Vec(1/((1-4*x)*(1-8*x)*(1-12*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

Formula

a(n) = (4^n)*Stirling2(n+3, 3), n >= 0, with Stirling2(n, m) = A008277(n, m).
a(n) = (4^n - 8*8^n + 9*12^n)/2.
G.f.: 1/((1-4*x)*(1-8*x)*(1-12*x)).
E.g.f.: (d^3/dx^3)((((exp(4*x)-1)/4)^3)/3!) = (exp(4*x) - 8*exp(8*x) + 9*exp(12*x))/2.
a(0)=1, a(1)=24, a(2)=400; for n > 2, a(n) = 24*a(n-1) - 176*a(n-2) + 384*a(n-3). - Vincenzo Librandi, Jul 03 2013
a(n) = 30*a(n-1) - 96*a(n-2) + 4^n. - Vincenzo Librandi, Jul 03 2013
Previous Showing 11-13 of 13 results.

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

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