A051232 -id:A051232 - 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)

A053116 a(n) = ((9n+10)(!^9))/10, related to A045756 ((9n+1)(!^9) 9-factorials).

Original entry on oeis.org

1, 19, 532, 19684, 905464, 49800520, 3187233280, 232668029440, 19078778414080, 1736168835681280, 173616883568128000, 18924240308925952000, 2233060356453262336000, 283598665269564316672000

Offset: 0

Wolfdieter Lang

Row m=10 of the array A(10; m,n) := ((9*n+m)(!^9))/m(!^9), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..327

Cf. A051232, A045756, A035012-3, A035017-8, A035020-3 (rows m=0..9).

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( 1/(1 - 9*x)^(19/9))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 26 2018

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 18, 3*5!, 9}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nmax = 50}, CoefficientList[Series[1/(1 - 9*x)^(19/9), {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Aug 26 2018 *)

x='x+O('x^25); Vec(serlaplace(1/(1 - 9*x)^(19/9))) \\ G. C. Greubel, Aug 26 2018

a(n) = ((9*n+10)(!^9))/10(!^9) = A045756(n+2)/10.

E.g.f.: 1/(1-9*x)^(19/9).

A147629 9-factorial numbers (4).

Original entry on oeis.org

1, 5, 70, 1610, 51520, 2112320, 105616000, 6231344000, 423731392000, 32627317184000, 2805949277824000, 266565181393280000, 27722778864901120000, 3132674011733826560000, 382186229431526840320000, 50066396055530016081920000, 7009295447774202251468800000

Offset: 1

Vladimir Joseph Stephan Orlovsky, Nov 08 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 1..300

Cf. A035012, A035013, A035017, A035018, A035020, A035022, A035023, A045756, A048994, A049211, A051232, A053116, A132393.

[Round(9^(n-1)*Gamma(n-1 +5/9)/Gamma(5/9)): n in [1..20]]; // G. C. Greubel, Dec 03 2019

seq(9^(n-1)*pochhammer(5/9, n-1), n = 1..20); # G. C. Greubel, Dec 03 2019

Table[9^(n-1)*Pochhammer[5/9, n-1], {n,20}] (* G. C. Greubel, Dec 03 2019 *)

vector(20, n, prod(j=0,n-2, 9*j+5) ) \\ G. C. Greubel, Dec 03 2019

[9^(n-1)*rising_factorial(5/9, n-1) for n in (1..20)] # G. C. Greubel, Dec 03 2019

a(n+1) = Sum_{k=0..n} A132393(n,k)*5^k*9^(n-k). - Philippe Deléham, Nov 09 2008
From R. J. Mathar, Nov 09 2008: (Start)
a(n) = a(n-1) + (4 + 9*(n-2))*a(n-1) = (9*n-13)*a(n-1).
a(n) = 9^(n-1)*Gamma(n-4/9)/Gamma(5/9).
G.f.: z*2F0(5/9,1; -; 9*z). (End)
a(n) = (-4)^n*Sum_{k=0..n} (9/4)^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
Sum_{n>=1} 1/a(n) = 1 + (e/9^4)^(1/9)*(Gamma(5/9) - Gamma(5/9, 1/9)). - Amiram Eldar, Dec 21 2022

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

A147630 a(1) = 1; for n>1, a(n) = Product_{k = 1..n-1} (9k - 3).

Original entry on oeis.org

1, 6, 90, 2160, 71280, 2993760, 152681760, 9160905600, 632102486400, 49303993939200, 4289447472710400, 411786957380198400, 43237630524920832000, 4929089879840974848000, 606278055220439906304000, 80028703289098067632128000, 11284047163762827536130048000

Offset: 1

Vladimir Joseph Stephan Orlovsky, Nov 08 2008

Original name was: 9-factorial numbers (5).

Cf. A147629, A049211, A051232, A045756, A035012, A035013, A035017, A035018, A035020, A035022, A035023, A053116.

Cf. A048994, A097188, A132393.

[Round(9^n*Gamma(n+6/9)/Gamma(6/9)): n in [0..20]]; // Vincenzo Librandi, Feb 21 2015

s=1;lst={s};Do[s+=n*s;AppendTo[lst,s],{n,5,2*5!,9}];lst
Table[Product[9k-3,{k,1,n-1}],{n,20}] (* Harvey P. Dale, Sep 01 2016 *)

a(n):=n!*sum(binomial(k,n-k-1)*3^k*(-1)^(n-k-1)*binomial(n+k-1,n-1),k,1,n-1)/n; /* Vladimir Kruchinin, Apr 01 2011 */

a(n) = n--; prod(k=1, n, 9*k-3); \\ Michel Marcus, Feb 28 2015

a(n+1) = Sum_{k, 0<=k<=n}A132393(n,k)*6^k*9^(n-k). - Philippe Deléham, Nov 09 2008
a(n) = n!*(Sum_{k=1..n-1} binomial(k,n-k-1)*3^k*(-1)^(n-k-1)*binomial(n+k-1,n-1))/n for n>1, also a(n) = n!*A097188(n-1). - Vladimir Kruchinin, Apr 01 2011
a(n) = (-3)^n*sum_{k=0..n} 3^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
a(n) = round(9^n * Gamma(n+6/9) / Gamma(6/9)). - Vincenzo Librandi, Feb 21 2015
Sum_{n>=1} 1/a(n) = 1 + (e/9^3)^(1/9)*(Gamma(2/3) - Gamma(2/3, 1/9)). - Amiram Eldar, Dec 21 2022

New name from Peter Bala, Feb 20 2015

More terms from Michel Marcus, Feb 28 2015

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)

A051231 Generalized Stirling number triangle of the first kind.

Original entry on oeis.org

1, -9, 1, 162, -27, 1, -4374, 891, -54, 1, 157464, -36450, 2835, -90, 1, -7085880, 1797714, -164025, 6885, -135, 1, 382637520, -104162436, 10655064, -535815, 14175, -189, 1, -24106163760, 6944870988, -775431468, 44411409, -1428840, 26082, -252, 1

Offset: 1

Wolfdieter Lang

T(n,m) = R_n^m(a=0, b=9) in the notation of the given 1962 reference.
T(n,m) is a Jabotinsky matrix, i.e., the monic row polynomials E(n,x) := Sum_{m=1..n} T(n,m)*x^m = Product_{j=0..n-1} (x - 9*j), n >= 1, and E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
From Petros Hadjicostas, Jun 06 2020: (Start)
For nonnegative integers n, m and complex numbers a, b (with b <> 0), the numbers R_n^m(a,b) were introduced by Mitrinovic (1961) using slightly different notation. They were further examined by Mitrinovic and Mitrinovic (1962). Special cases were tabulated in this and other related papers.
Special cases of these numbers are related to numbers introduced by Nörlund (1924).
These numbers are defined via the g.f. Product_{r=0..n-1} (x - (a + b*r)) = Sum_{m=0..n} R_n^m(a,b)*x^m for n >= 0. As a result, R_n^m(a,b) = R_{n-1}^{m-1}(a,b) - (a + b*(n-1))*R_{n-1}^m(a,b) for n >= m >= 1 with R_1^0(a,b) = a, R_1^1(a,b) = 1, and R_n^m(a,b) = 0 for n < m. (Because an empty product is by definition 1, we may let R_0^0(a,b) = 1.)
With a = 0 and b = 1, we get the Stirling numbers of the first kind S1(n,m) = R_n^m(a=0, b=1) = A048994(n,m) which satisfy Product_{r=0}^{n-1} (x - r) = Sum_{m=0..n} S1(n,m)*x^m with S1(n,n) = 1 for n >= 0, S1(n,0) = 0 for n >= 1, and S1(n, m) = 0 for m > n. (Array A008275 is the same as array A048994 but with no zero row and no zero column.)
We have R_n^m(a,b) = Sum_{k=0}^{n-m} (-1)^k * a^k * b^(n-m-k) * binomial(m+k, k) * S1(n, m+k) for n >= m >= 0.
For the current array, T(n,m) = R_n^m(a=0, b=9) but with no zero row or column. (End)

			Triangle T(n,m) (with rows n >= 1 and columns m = 1..n) begins:
         1;
        -9,       1;
       162,     -27,       1;
     -4374,     891,     -54,    1;
    157464,  -36450,    2835,  -90,    1;
  -7085880, 1797714, -164025, 6885, -135, 1;
   ...
3rd row o.g.f.: E(3,x) = Product_{j=0..2} (x-9*j) = 162*x - 27*x^2 + x^3. [Edited by _Petros Hadjicostas_, Jun 06 2020]

D. S. Mitrinovic, Sur une relation de récurrence relative aux nombres de Bernoulli, Comptes rendus de l'Académie des sciences de Paris, t. 250 (1960), 4266-4267.
D. S. Mitrinovic, Sur une classe de nombres reliés aux nombres de Stirling, Comptes rendus de l'Académie des sciences de Paris, t. 252 (1961), 2354-2356.
D. S. Mitrinovic and R. S. Mitrinovic, Sur les polynômes de Stirling, Bulletin de la Société des mathématiciens et physiciens de la R. P. de Serbie, t. 10 (1958), 43-49.
D. S. Mitrinovic and R. S. Mitrinovic, Sur les nombres de Stirling et les nombres de Bernoulli d'ordre supérieux, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 43 (1960), 1-63.
D. S. Mitrinovic and R. S. Mitrinovic, Sur une classe de nombres se rattachant aux nombres de Stirling--Appendice: Table des nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 60 (1961), 1-15 and 17-62.
D. S. Mitrinovic and R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 77 (1962), 1-77.
D. S. Mitrinovic and R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 77 (1962), 1-77 [jstor stable version].
D. S. Mitrinovic and R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling. V, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 132/142 (1965), 1-22.
Niels Nörlund, Vorlesungen über Differenzenrechnung, Springer, Berlin, 1924.
Wikipedia, Dragoslav Mitrinovic.

First (m=1) column sequence is A051232(n-1).
Row sums (signed triangle): A049211(n-1)*(-1)^(n-1).
Row sums (unsigned triangle): A045756(n).
Cf. A008275 (b=1 triangle), A048994 (b=1 triangle), A051187 (b=8 triangle).

T(n, m) = T(n-1, m-1) - 9*(n-1)*T(n-1, m), n >= m >= 1; T(n, m) := 0, n < m; T(n, 0) := 0 for n >= 1; T(0, 0) = 1.
E.g.f. for the m-th column of the signed triangle: (log(1 + 9*x)/9)^m/m!.
From Petros Hadjicostas, Jun 07 2020: (Start)
T(n,m) = 9^(n-m)*Stirling1(n,m) = 9^(n-m)*A048994(n,m) = 9^(n-m)*A008275(n,m) for n >= m >= 1.
Bivariate e.g.f.-o.g.f.: Sum_{n,m >= 1} T(n,m)*x^n*y^m/n! = exp((y/9)*log(1 + 9*x)) - 1 = (1 + 9*x)^(y/9) - 1. (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)

A147631 9-factorial numbers (6).

Original entry on oeis.org

1, 7, 112, 2800, 95200, 4093600, 212867200, 12984899200, 908942944000, 71806492576000, 6318971346688000, 612940220628736000, 64971663386646016000, 7471741289464291840000, 926495919893572188160000, 123223957345845101025280000, 17497801943110004345589760000

Offset: 1

Vladimir Joseph Stephan Orlovsky, Nov 08 2008

nonn

Cf. A147630, A147629, A049211, A051232, A045756, A035012, A035013, A035017, A035018, A035020, A035022, A035023, A053116.

Cf. A048994, A132393.

s=1;lst={s};Do[s+=n*s;AppendTo[lst,s],{n,6,2*5!,9}];lst

a(n+1) = Sum_{k=0..n} A132393(n,k)*7^k*9^(n-k). - Philippe Deléham, Nov 09 2008
a(n) = (-2)^n*Sum_{k=0..n} (9/2)^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
Sum_{n>=1} 1/a(n) = 1 + (e/9^2)^(1/9)*(Gamma(7/9) - Gamma(7/9, 1/9)). - Amiram Eldar, Dec 21 2022

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A051262 10-factorial numbers.

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A053116 a(n) = ((9*n+10)(!^9))/10, related to A045756 ((9*n+1)(!^9) 9-factorials).

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A147629 9-factorial numbers (4).

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

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A147630 a(1) = 1; for n>1, a(n) = Product_{k = 1..n-1} (9k - 3).

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

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A051231 Generalized Stirling number triangle of the first kind.

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A053116 a(n) = ((9n+10)(!^9))/10, related to A045756 ((9n+1)(!^9) 9-factorials).