A051189 -id:A051189 - cp's OEIS frontend

A153511 a(n) = 4 * A051189(n).

4, 32, 512, 12288, 393216, 15728640, 754974720, 42278584320, 2705829396480, 194819716546560, 15585577323724800, 1371530804487782400, 131666957230827110400, 13693363552006019481600

Offset: 0

Roger L. Bagula, Dec 28 2008

nonn

A binomial sequence that produces Pi: 1/Pi= Binomial[2*n+1,n+1/2]/(2*n+1)!!

G. C. Greubel, Table of n, a(n) for n = 0..100

Cf. A051189.

Table[(2*n + 1)!!*Pi*Gamma[2*n + 2]/(Gamma[n + 3/2]^2), {n, 0, 20}]

a(n) = 4*n!*8^n; \\ Michel Marcus, Aug 22 2016

a(n) = 4 * A051189(n).
From Ilya Gutkovskiy, Aug 22 2016: (Start)
E.g.f.: 4/(1 - 8*x).
a(n) ~ sqrt(Pi)*2^(3*n+5/2)*n^(n+1/2)/exp(n). (End)

A256890 Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1), where f(x) = x + 2.

Original entry on oeis.org

1, 2, 2, 4, 12, 4, 8, 52, 52, 8, 16, 196, 416, 196, 16, 32, 684, 2644, 2644, 684, 32, 64, 2276, 14680, 26440, 14680, 2276, 64, 128, 7340, 74652, 220280, 220280, 74652, 7340, 128, 256, 23172, 357328, 1623964, 2643360, 1623964, 357328, 23172, 256, 512, 72076, 1637860, 10978444, 27227908, 27227908, 10978444, 1637860, 72076, 512

Offset: 0

Dale Gerdemann, Apr 12 2015

Related triangles may be found by varying the function f(x). If f(x) is a linear function, it can be parameterized as f(x) = a*x + b. With different values for a and b, the following triangles are obtained:
  a\b 1.......2.......3.......4.......5.......6
  -1  A144431
  0   A007318 A038208 A038221
  1   A008292 A256890 A257180 A257606 A257607
  2   A060187 A257609 A257611 A257613 A257615
  3   A142458 A257610 A257620 A257622 A257624 A257626
  4   A142459 A257612 A257621
  5   A142460 A257614 A257623
  6   A142461 A257616 A257625
  7   A142462 A257617 A257627
  8   A167884 A257618
  9   A257608 A257619
The row sums of these, and similarly constructed number triangles, are shown in the following table:
  a\b 1.......2.......3.......4.......5.......6.......7.......8.......9
  0   A000079 A000302 A000400
  1   A000142 A001715 A001725 A049388 A049198
  2   A000165 A002866 A002866 A051580 A051582
  3   A008544 A051578 A037559 A051605 A051607 A051609
  4   A001813 A047053 A000407 A034177 A051618 A051620 A051622
  5   A047055 A008546 A008548 A034300 A034325 A051688 A051690
  6   A047657 A049308 A047058 A034689 A034724 A034788 A053101 A053103
  7   A084947 A144827 A049209 A045754 A034830 A034832 A034834 A053105
  8   A084948 A144828 A147626 A051189 A034908 A034910 A034912 A034976 A053115
  9   A084949 A144829 A147630 A049211 A045756 A035013 A035018 A035021 A035023
  10                                  A051262 A035265 A035273         A035277
  11                                  A254322
  12                                          A145448
The formula can be further generalized to: t(n,m) = f(m+s)*t(n-1,m) + f(n-s)*t(n,m-1), where f(x) = a*x + b. The following table specifies triangles with nonzero values for s (given after the slash).
  a\b  0           1           2          3
  -2   A130595/1
  -1
  0
  1    A110555/-1  A120434/-1  A144697/1  A144699/2
With the absolute value, f(x) = |x|, one obtains A038221/3, A038234/4, A038247/5, A038260/6, A038273/7, A038286/8, A038299/9 (with value for s after the slash).
If f(x) = A000045(x) (Fibonacci) and s = 1, the result is A010048 (Fibonomial).
In the notation of Carlitz and Scoville, this is the triangle of generalized Eulerian numbers A(r, s | alpha, beta) with alpha = beta = 2. Also the array A(2,1,4) in the notation of Hwang et al. (see page 31). - Peter Bala, Dec 27 2019

			Array, t(n, k), begins as:
   1,    2,      4,        8,        16,         32,          64, ...;
   2,   12,     52,      196,       684,       2276,        7340, ...;
   4,   52,    416,     2644,     14680,      74652,      357328, ...;
   8,  196,   2644,    26440,    220280,    1623964,    10978444, ...;
  16,  684,  14680,   220280,   2643360,   27227908,   251195000, ...;
  32, 2276,  74652,  1623964,  27227908,  381190712,  4677894984, ...;
  64, 7340, 357328, 10978444, 251195000, 4677894984, 74846319744, ...;
Triangle, T(n, k), begins as:
    1;
    2,     2;
    4,    12,      4;
    8,    52,     52,       8;
   16,   196,    416,     196,      16;
   32,   684,   2644,    2644,     684,      32;
   64,  2276,  14680,   26440,   14680,    2276,     64;
  128,  7340,  74652,  220280,  220280,   74652,   7340,   128;
  256, 23172, 357328, 1623964, 2643360, 1623964, 357328, 23172,   256;

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened.)
L. Carlitz and R. Scoville, Generalized Eulerian numbers: combinatorial applications, J. für die reine und angewandte Mathematik, 265 (1974): 110-37. See Section 3.
Dale Gerdemann, A256890, Plot of t(m,n) mod k , YouTube, 2015.
Hsien-Kuei Hwang, Hua-Huai Chern, and Guan-Huei Duh, An asymptotic distribution theory for Eulerian recurrences with applications, arXiv:1807.01412 [math.CO], 2018-2019.

Cf. A000079, A001715, A008292, A038208, A257180, A257606, A257607, A257609.

Cf. A257610, A257612, A257614, A257616, A257617, A257618, A257619.

A256890:= func< n,k | (&+[(-1)^(k-j)*Binomial(j+3,j)*Binomial(n+4,k-j)*(j+2)^n: j in [0..k]]) >;
[A256890(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Oct 18 2022

Table[Sum[(-1)^(k-j)*Binomial[j+3, j] Binomial[n+4, k-j] (j+2)^n, {j,0,k}], {n,0, 9}, {k,0,n}]//Flatten (* Michael De Vlieger, Dec 27 2019 *)

t(n,m) = if ((n<0) || (m<0), 0, if ((n==0) && (m==0), 1, (m+2)*t(n-1, m) + (n+2)*t(n, m-1)));
tabl(nn) = {for (n=0, nn, for (k=0, n, print1(t(n-k, k), ", ");); print(););} \\ Michel Marcus, Apr 14 2015

def A256890(n,k): return sum((-1)^(k-j)*Binomial(j+3,j)*Binomial(n+4,k-j)*(j+2)^n for j in range(k+1))
flatten([[A256890(n,k) for k in range(n+1)] for n in range(11)]) # G. C. Greubel, Oct 18 2022

T(n,k) = t(n-k, k); t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0 else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 2.
Sum_{k=0..n} T(n, k) = A001715(n).
T(n,k) = Sum_{j = 0..k} (-1)^(k-j)*binomial(j+3,j)*binomial(n+4,k-j)*(j+2)^n. - Peter Bala, Dec 27 2019
Modified rule of Pascal: T(0,0) = 1, T(n,k) = 0 if k < 0 or k > n else T(n,k) = f(n-k) * T(n-1,k-1) + f(k) * T(n-1,k), where f(x) = x + 2. - Georg Fischer, Nov 11 2021
From G. C. Greubel, Oct 18 2022: (Start)
T(n, n-k) = T(n, k).
T(n, 0) = A000079(n). (End)

A051232 9-factorial numbers.

Original entry on oeis.org

1, 9, 162, 4374, 157464, 7085880, 382637520, 24106163760, 1735643790720, 140587147048320, 12652843234348800, 1252631480200531200, 135284199861657369600, 15828251383813912243200, 1994359674360552942643200, 269238556038674647256832000

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_9)^n. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 07 2001
a(n) = 9*A035023(n) = Product_{k=1..n} 9*k, n >= 1; a(0) := 1.
Pi^n/a(n) is the volume of a 2n-dimensional ball with radius 1/3. - Peter Luschny, Jul 24 2012

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

Cf. A047058, A051188, A051189. a(n) = A051231(n-1, 0), A053116 (first column of triangle).

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

with(combstruct):A:=[N,{N=Cycle(Union(Z$9))},labeled]: seq(count(A,size=n+1)/9, n=0..14); # Zerinvary Lajos, Dec 05 2007

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 8, 2*5!, 9}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)

a(n) = n!*9^n =: (9*n)(!^9).
E.g.f.: 1/(1-9*x).
G.f.: 1/(1 - 9*x/(1 - 9*x/(1 - 18*x/(1 - 18*x/(1 - 27*x/(1 - 27*x/(1 - ...))))))), a continued fraction. - Ilya Gutkovskiy, Aug 09 2017
From Amiram Eldar, Jun 25 2020: (Start)
Sum_{n>=0} 1/a(n) = e^(1/9).
Sum_{n>=0} (-1)^n/a(n) = e^(-1/9). (End)

A034976 One eighth of octo-factorial numbers.

Original entry on oeis.org

1, 16, 384, 12288, 491520, 23592960, 1321205760, 84557168640, 6088116142080, 487049291366400, 42860337640243200, 4114592413463347200, 427917611000188108800, 47926772432021068185600, 5751212691842528182272000, 736155224555843607330816000

Offset: 1

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 1..325
Index entries for sequences related to factorial numbers.

Cf. A034908, A034909, A034910, A034911, A034912, A034975, A045755, A051189.

[8^(n-1)*Factorial(n): n in [1..40]]; // G. C. Greubel, Oct 20 2022

Table[8^(n-1)*n!, {n,40}] (* G. C. Greubel, Oct 20 2022 *)

[8^(n-1)*factorial(n) for n in range(1,40)] # G. C. Greubel, Oct 20 2022

8*a(n) = (8*n)!^8 = Product_{j=1..n} 8*j = 8^n*n!.
E.g.f.: (-1+(1-8*x)^(-1))/8.
G.f.: x/(1-16*x/(1-8*x/(1-24*x/(1-16*x/(1-32*x/(1-24*x/(1-40*x/(1-32*x/(1-... (continued fraction). - Philippe Deléham, Jan 07 2012
From Amiram Eldar, Jan 08 2022: (Start)
Sum_{n>=1} 1/a(n) = 8*(exp(1/8)-1).
Sum_{n>=1} (-1)^(n+1)/a(n) = 8*(1-exp(-1/8)). (End)

A051187 Generalized Stirling number triangle of the first kind.

Original entry on oeis.org

1, -8, 1, 128, -24, 1, -3072, 704, -48, 1, 98304, -25600, 2240, -80, 1, -3932160, 1122304, -115200, 5440, -120, 1, 188743680, -57802752, 6651904, -376320, 11200, -168, 1, -10569646080, 3425697792, -430309376, 27725824, -1003520, 20608, -224, 1

Offset: 1

Wolfdieter Lang

T(n,m)= R_n^m(a=0, b=8) 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 - 8*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 07 2020: (Start)
For integers n, m >= 0 and complex numbers a, b (with b <> 0), the numbers R_n^m(a,b) were introduced by Mitrinovic (1961) and further examined by Mitrinovic and Mitrinovic (1962). Such numbers are related to the work of Nörlund (1924).
They are defined via 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, R_n^m(a,b) = 0 for n < m, and 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).
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=8) but with no zero row or column. (End)

			Triangle T(n,m) (with rows n >= 1 and columns m = 1..n) begins:
          1;
         -8,         1;
        128,       -24,       1;
      -3072,       704,     -48,       1;
      98304,    -25600,    2240,     -80,     1;
   -3932160,   1122304, -115200,    5440,  -120,    1;
  188743680, -57802752, 6651904, -376320, 11200, -168, 1;
  ...
3rd row o.g.f.: E(3,x) = Product_{j=0..2} (x - 8*j) = 128*x - 24*x^2 + x^3.

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 supérieur, 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].
Niels Nörlund, Vorlesungen über Differenzenrechnung, Springer, Berlin, 1924.

First (m=1) column sequence is: A051189(n-1).
Row sums (signed triangle): A049210(n-1)*(-1)^(n-1).
Row sums (unsigned triangle): A045755(n).
The b=1..7 triangles are: A008275 (Stirling1 triangle), A039683, A051141, A051142, A051150, A051151, A051186.

T(n, m) = T(n-1, m-1) - 8*(n-1)*T(n-1, m) for n >= m >= 1; T(n, m) := 0 for 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 + 8*x)/8)^m/m!.
From Petros Hadjicostas, Jun 07 2020: (Start)
T(n,m) = 8^(n-m)*Stirling1(n,m) = 8^(n-m)*A048994(n,m) = 8^(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/8)*log(1 + 8*x)) - 1 = (1 + 8*x)^(y/8) - 1. (End)

A051262 10-factorial numbers.

Original entry on oeis.org

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)

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

A053115 a(n) = ((8n+10)(!^8))/20, related to A034908 ((8n+2)(!^8) octo- or 8-factorials).

Original entry on oeis.org

1, 18, 468, 15912, 668304, 33415200, 1938081600, 127913385600, 9465590534400, 776178423820800, 69856058143872000, 6845893698099456000, 725664731998542336000, 82725779447833826304000

Offset: 0

Wolfdieter Lang

Row m=10 of the array A(9; m,n) := ((8*n+m)(!^8))/m(!^8), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..332

Cf. A051189, A045755, A034908-12, A034975-6, A053114 (rows m=0..9).

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-8*x)^(9/4))); [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, 17, 5!, 8}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nmax = 50}, CoefficientList[Series[1/(1 - 8*x)^(9/4), {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Aug 26 2018 *)

x='x+O('x^25); Vec(serlaplace(1/(1-8*x)^(9/4))) \\ G. C. Greubel, Aug 26 2018

a(n) = ((8*n+10)(!^8))/10(!^8) = A034908(n+2)/10.
E.g.f.: 1/(1-8*x)^(9/4).
G.f.: 1/(1-18x/(1-8x/(1-26x/(1-16x/(1-34x/(1-24x/(1-42x/(1-32x/(1-... (continued fraction). - Philippe Deléham, Jan 07 2012

A098432 Coefficients of polynomials S(n,x) related to Springer numbers.

Original entry on oeis.org

1, 8, 7, 128, 304, 177, 3072, 13952, 21080, 10199, 98304, 724992, 2016000, 2441056, 1051745, 3932160, 42762240, 187643904, 407505664, 428605352, 169913511, 188743680, 2839019520, 17974591488, 60428242944, 111985428352

Offset: 0

Ralf Stephan, Sep 07 2004

			S(0,x) = 1,
S(1,x) = 8*x + 7,
S(2,x) = 128*x^2 + 304*x + 177,
S(3,x) = 3072*x^3 + 13952*x^2 + 21080*x + 10199.

A. Randrianarivony and J. Zeng, Une famille des polynomes qui interpole plusieurs suites..., Adv. Appl. Math. 17 (1996), 1-26.

Cf. A001586. S(n, 1/2) = A000464(n+1), S(n, -1/2) = A000281(n).
Leading coefficients are A051189. Constant terms are in A098433.
Cf. A001586. S(n, 1/2) = A000464(n), S(n, -1/2) = A000281(n).

S(n,x)=if(n<1,1,(2*x+2)*(2*x+4)*S(n-1,x+2)-(2*x+1)^2*S(n-1,x))

Recurrence: S(0, x)=1, S(n, x)=(2x+2)(2x+4)S(n-1, x+2)-(2x+1)^2S(n-1, x).

G.f.: Sum[n>=0, S(n, x)t^n] = 1/(1+t-4*2(x+1)t/(1-4*2(x+2)t/(1+t-4*4(x+3)t/(1-4+4(x+4)t/...)))).

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)

cp's OEIS Frontend

A153511 a(n) = 4 * A051189(n).

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A256890 Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 2.

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A051232 9-factorial numbers.

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A034976 One eighth of octo-factorial numbers.

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A051187 Generalized Stirling number triangle of the first kind.

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A051262 10-factorial numbers.

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

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A256890 Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1), where f(x) = x + 2.

A053115 a(n) = ((8n+10)(!^8))/20, related to A034908 ((8n+2)(!^8) octo- or 8-factorials).