A051188 -id:A051188 - cp's OEIS frontend

A051262 10-factorial numbers.

1, 10, 200, 6000, 240000, 12000000, 720000000, 50400000000, 4032000000000, 362880000000000, 36288000000000000, 3991680000000000000, 479001600000000000000, 62270208000000000000000

Offset: 0

Wolfdieter Lang

For n >= 1 a(n) is the order of the wreath product of the symmetric group S_n and the Abelian group (C_10)^n. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 07 2001

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for sequences related to factorial numbers

a(n) = A048176(n+1, 0)*(-1)^n (first column of unsigned triangle).

Cf. A047058, A051188, A051189, A051232, A035279.

[10^n*Factorial(n): n in [0..20]]; // Vincenzo Librandi, Oct 05 2011

with(combstruct):A:=[N,{N=Cycle(Union(Z$10))},labeled]: seq(count(A,size=n)/10,n=0..14); # Zerinvary Lajos, Dec 05 2007

Array[#!*10^# &, 14, 0] (* Michael De Vlieger, Sep 04 2017 *)

a(n) = 10*A035279(n) = Product_{k=1..n} 10*k, n >= 1; a(0) := 1.
a(n) = n!*10^n =: (10*n)(!^10);
E.g.f.: 1/(1-10*x).
G.f.: 1/(1 - 10*x/(1 - 10*x/(1 - 20*x/(1 - 20*x/(1 - 30*x/(1 - 30*x/(1 - ...))))))), a continued fraction. - Ilya Gutkovskiy, May 12 2017
From Amiram Eldar, Jun 25 2020: (Start)
Sum_{n>=0} 1/a(n) = e^(1/10).
Sum_{n>=0} (-1)^n/a(n) = e^(-1/10). (End)

A053106 a(n) = ((7n+10)(!^7))/10(1^7), related to A034830 (((7n+3)(!^7))/3 sept-, or 7-factorials).

Original entry on oeis.org

1, 17, 408, 12648, 480624, 21628080, 1124660160, 66354949440, 4379426663040, 319698146401920, 25575851712153600, 2225099098957363200, 209159315301992140800, 21125090845501206220800

Offset: 0

Wolfdieter Lang

Row m=10 of the array A(8; m,n) := ((7*n+m)(!^7))/m(!^7), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..337

Cf. A051188, A045754(n+1), A034829-A034834(n+1), A053104-A053106 (rows m=0..10).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-7*x)^(17/7))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 16 2018

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 16, 5!, 7}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 7*x)^(17/7), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 16 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1-7*x)^(17/7))) \\ G. C. Greubel, Aug 16 2018

a(n) = ((7*n+10)(!^7))/10(!^7) = A034830(n+2)/10.

E.g.f.: 1/(1-7*x)^(17/7).

A196347 Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).

Original entry on oeis.org

1, 1, 1, 2, 4, 2, 6, 18, 18, 6, 24, 96, 144, 96, 24, 120, 600, 1200, 1200, 600, 120, 720, 4320, 10800, 14400, 10800, 4320, 720, 5040, 35280, 105840, 176400, 176400, 105840, 35280, 5040, 40320, 322560, 1128960, 2257920, 2822400, 2257920, 1128960, 322560, 40320

Offset: 0

Philippe Deléham, Oct 28 2011

Unsigned version of A021012.

Equal to A136572*A007318.

			Triangle begins:
    1;
    1,   1;
    2,   4,    2;
    6,  18,   18,    6;
   24,  96,  144,   96,  24;
  120, 600, 1200, 1200, 600, 120;
  ...

G. C. Greubel, Table of n, a(n) for n = 0..495
P. Bala, Deformations of the Hadamard product of power series
Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021.
M. Dukes, C. D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.

Cf. A007318, A008619, A021012, A084938, A136572.

Columns include: A000142, A001563, A001804, A001805, A001806, A001807.

/* As triangle */ [[Factorial(n)*Binomial(n, k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Sep 28 2015

Table[n!*Binomial[n, j], {n, 0, 30}, {j, 0, n}] (* G. C. Greubel, Sep 27 2015 *)

factorial(n)*binomial(n,k) # Danny Rorabaugh, Sep 27 2015

T(n,k) is given by (1,1,2,2,3,3,4,4,5,5,6,6,...) DELTA (1,1,2,2,3,3,4,4,5,5,6,6, ...) where DELTA is the operator defined in A084938.
Sum_{k>=0} T(m,k)*T(n,k) = (m+n)!.
T(2n,n) = A122747(n).
Sum_{k>=0} T(n,k)^2 = A010050(n) = (2n)!.
Sum_{k>=0} T(n,k)*x^k = A000007(n), A000142(n), A000165(n), A032031(n), A047053(n), A052562(n), A047058(n), A051188(n), A051189(n), A051232(n), A051262(n), A196258(n), A145448(n) for x = -1,0,1,2,3,4,5,6,7,8,9,10,11 respectively.
The row polynomials have the form (x + 1) o (x + 2) o ... o (x + n), where o denotes the black diamond multiplication operator of Dukes and White. See example E10 in the Bala link. - Peter Bala, Jan 18 2018

Name exchanged with a formula by Peter Luschny, Feb 01 2015

A053104 a(n) = ((7n+8)(!^7))/8, related to A045754 ((7n+1)(!^7) sept-, or 7-factorials).

Original entry on oeis.org

1, 15, 330, 9570, 344520, 14814360, 740718000, 42220926000, 2702139264000, 191851887744000, 14964447244032000, 1271978015742720000, 117021977448330240000, 11585175767384693760000

Offset: 0

Wolfdieter Lang

Row m=8 of the array A(8; m,n) := ((7*n+m)(!^7))/m(!^7), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..337

Cf. A051188, A045754(n+1), A034829-34(n+1), A053104-A053106 (rows m=0..10).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-7*x)^(15/7))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 16 2018

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 14, 5!, 7}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 7*x)^(15/7), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 16 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1-7*x)^(15/7))) \\ G. C. Greubel, Aug 16 2018

a(n) = ((7*n+8)(!^7))/8(!^7) = A045754(n+2)/8.

E.g.f.: 1/(1-7*x)^(15/7).

A053105 a(n) = ((7n+9)(!^7))/9(!^7), related to A034829 (((7n+2)(!^7))/2 sept-, or 7-factorials).

Original entry on oeis.org

1, 16, 368, 11040, 408480, 17973120, 916629120, 53164488960, 3455691782400, 248809808332800, 19655974858291200, 1690413837813043200, 157208486916613017600, 15720848691661301760000

Offset: 0

Wolfdieter Lang

Row m=9 of the array A(8; m,n) := ((7*n+m)(!^7))/m(!^7), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..337

Cf. A051188, A045754(n+1), A034829-34(n+1), A053104-A053106 (rows m=0..10).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-7*x)^(16/7))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 16 2018

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 15, 5!, 7}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
CoefficientList[Series[1/(1-7x)^(16/7),{x,0,20}],x]Range[0,20]! (* Harvey P. Dale, Sep 11 2011 *)

x='x+O('x^30); Vec(serlaplace(1/(1-7*x)^(16/7))) \\ G. C. Greubel, Aug 16 2018

a(n) = ((7*n+9)(!^7))/9(!^7)= A034829(n+2)/9.

E.g.f.: 1/(1-7*x)^(16/7).

A131182 Table T(n,k) = n!*k^n, read by upwards antidiagonals.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 6, 8, 3, 1, 0, 24, 48, 18, 4, 1, 0, 120, 384, 162, 32, 5, 1, 0, 720, 3840, 1944, 384, 50, 6, 1, 0, 5040, 46080, 29160, 6144, 750, 72, 7, 1, 0, 40320, 645120, 524880, 122880, 15000, 1296, 98, 8, 1, 0, 362880, 10321920, 11022480, 2949120, 375000, 31104, 2058, 128, 9, 1

Offset: 0

Philippe Deléham, Sep 25 2007

For k>0, T(n,k) is the n-th moment of the exponential distribution with mean = k. - Geoffrey Critzer, Jan 06 2019

T(n,k) is the minimum value of Product_{i=1..n} Sum_{j=1..k} r_j[i] where each r_j is a permutation of {1..n}. For the maximum value, see A331988. - Chai Wah Wu, Sep 01 2022

			The (inverted) table begins:
k=0: 1, 0,   0,    0,      0,       0, ... (A000007)
k=1: 1, 1,   2,    6,     24,     120, ... (A000142)
k=2: 1, 2,   8,   48,    384,    3840, ... (A000165)
k=3: 1, 3,  18,  162,   1944,   29160, ... (A032031)
k=4: 1, 4,  32,  384,   6144,  122880, ... (A047053)
k=5: 1, 5,  50,  750,  15000,  375000, ... (A052562)
k=6: 1, 6,  72, 1296,  31104,  933120, ... (A047058)
k=7: 1, 7,  98, 2058,  57624, 2016840, ... (A051188)
k=8: 1, 8, 128, 3072,  98304, 3932160, ... (A051189)
k=9: 1, 9, 162, 4374, 157464, 7085880, ... (A051232)
Main diagonal is 1, 1, 8, 162, 6144, 375000, ... (A061711).

Chai Wah Wu, Permutations r_j such that ∑i∏j r_j(i) is maximized or minimized, arXiv:1508.02934 [math.CO], 2015-2020.
Chai Wah Wu, On rearrangement inequalities for multiple sequences, arXiv:2002.10514 [math.CO], 2020.

Columns k=0-9 give: A000007, A000142, A000165, A032031, A047053, A052562, A047058, A051188, A051189, A051232.
Main diagonal gives A061711.
Cf. A292783, A303489, A331988.

T:= (n,k)-> n!*k^n:
seq(seq(T(d-k, k), k=0..d), d=0..12);  # Alois P. Heinz, Jan 06 2019

from math import factorial
def A131182_T(n, k): # compute T(n, k)
    return factorial(n)*k**n # Chai Wah Wu, Sep 01 2022

From Ilya Gutkovskiy, Aug 11 2017: (Start)
G.f. of column k: 1/(1 - k*x/(1 - k*x/(1 - 2*k*x/(1 - 2*k*x/(1 - 3*k*x/(1 - 3*k*x/(1 - ...))))))), a continued fraction.
E.g.f. of column k: 1/(1 - k*x). (End)

A196258 a(n) = 11^n*n!.

Original entry on oeis.org

1, 11, 242, 7986, 351384, 19326120, 1275523920, 98215341840, 8642950081920, 855652058110080, 94121726392108800, 11388728893445164800, 1503312213934761753600, 214973646592670930764800, 33105941575271323337779200

Offset: 0

Philippe Deléham, Oct 27 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A000142, A000165, A032031, A047053, A052562, A047058, A051188, A051189, A051232, A051262, A145448.

[(Factorial(n)*11^n): n in [0..20]]; // Vincenzo Librandi, Oct 28 2011

Table[11^n n!,{n,0,20}] (* Harvey P. Dale, Sep 21 2015 *)

a(n)=11^n*n! \\ Charles R Greathouse IV, Nov 22 2011

a(n) = 11^n*n!.
E.g.f.: 1/(1-11*x).
From Amiram Eldar, Jun 25 2020: (Start)
Sum_{n>=0} 1/a(n) = e^(1/11).
Sum_{n>=0} (-1)^n/a(n) = e^(-1/11). (End)

cp's OEIS Frontend

A051262 10-factorial numbers.

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A053106 a(n) = ((7*n+10)(!^7))/10(1^7), related to A034830 (((7*n+3)(!^7))/3 sept-, or 7-factorials).

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A196347 Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).

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A053104 a(n) = ((7*n+8)(!^7))/8, related to A045754 ((7*n+1)(!^7) sept-, or 7-factorials).

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A053105 a(n) = ((7*n+9)(!^7))/9(!^7), related to A034829 (((7*n+2)(!^7))/2 sept-, or 7-factorials).

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A131182 Table T(n,k) = n!*k^n, read by upwards antidiagonals.

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A196258 a(n) = 11^n*n!.

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A053106 a(n) = ((7n+10)(!^7))/10(1^7), related to A034830 (((7n+3)(!^7))/3 sept-, or 7-factorials).

A053104 a(n) = ((7n+8)(!^7))/8, related to A045754 ((7n+1)(!^7) sept-, or 7-factorials).

A053105 a(n) = ((7n+9)(!^7))/9(!^7), related to A034829 (((7n+2)(!^7))/2 sept-, or 7-factorials).