A075498 -id:A075498 - cp's OEIS frontend

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

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

Peter Luschny, Dec 30 2015

			[ 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]

Peter Luschny, The Bell transform
Richell O. Celeste, Roberto B. Corcino, and Ken Joffaniel M. Gonzales. Two Approaches to Normal Order Coefficients, Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.

Cf. A007696, A264428, A264429.

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

# 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

Peter Luschny, Dec 30 2015

			[ 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]

Peter Luschny, The Bell transform
Richell O. Celeste, Roberto B. Corcino, and Ken Joffaniel M. Gonzales, Two Approaches to Normal Order Coefficients, Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.

Cf. A007559, A264428, A264429.

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

# 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))

A028085 Expansion of 1/((1-3x)(1-6x)(1-9x)(1-12x)).

Original entry on oeis.org

1, 30, 585, 9450, 137781, 1888110, 24862545, 318755250, 4012058061, 49847787990, 613622150505, 7503229474650, 91300979746341, 1106997911204670, 13386607046238465, 161563913916523650

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (30,-315,1350,-1944).

Fourth column of triangle A075498.

Cf. A017933, A075515.

CoefficientList[Series[1/((1-3x)(1-6x)(1-9x)(1-12x)),{x,0,30}],x] (* or *) LinearRecurrence[{30,-315,1350,-1944},{1,30,585,9450},30] (* Harvey P. Dale, Feb 06 2015 *)

Vec(1/((1-3*x)*(1-6*x)*(1-9*x)*(1-12*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = (3^n)*Stirling2(n+4, 4), n >= 0, with Stirling2(n, m) = A008277(n, m).
a(n) = Sum_{m=0..3} (A075513(4, m)*((m+1)*3)^n)/3!.
G.f.: 1/Product_{k=1..4} (1-3*k*x).
E.g.f.: (d^4/dx^4)((((exp(3*x)-1)/3)^4)/4!) = Sum_{m=0..3} (A075513(4, m)*exp(3*(m+1)*x))/3!.
a(n) = (12^(n+3) - 3*9^(n+3) + 3*6^(n+3) - 3^(n+3))/162. - Yahia Kahloune, Jun 10 2013
a(0)=1, a(1)=30, a(2)=585, a(3)=9450, a(n) = 30*a(n-1) - 315*a(n-2) + 1350*a(n-3) - 1944*a(n-4). - Harvey P. Dale, Feb 06 2015

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Sage

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Sage

A028085 Expansion of 1/((1-3x)(1-6x)(1-9x)(1-12x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula