A008546 -id:A008546 - cp's OEIS frontend

A132393 Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.

1, 0, 1, 0, 1, 1, 0, 2, 3, 1, 0, 6, 11, 6, 1, 0, 24, 50, 35, 10, 1, 0, 120, 274, 225, 85, 15, 1, 0, 720, 1764, 1624, 735, 175, 21, 1, 0, 5040, 13068, 13132, 6769, 1960, 322, 28, 1, 0, 40320, 109584, 118124, 67284, 22449, 4536, 546, 36, 1, 0, 362880, 1026576, 1172700, 723680, 269325, 63273, 9450, 870, 45, 1

Offset: 0

Philippe Deléham, Nov 10 2007, Oct 15 2008, Oct 17 2008

Another name: Triangle of signless Stirling numbers of the first kind.
Triangle T(n,k), 0<=k<=n, read by rows given by [0,1,1,2,2,3,3,4,4,5,5,...] DELTA [1,0,1,0,1,0,1,0,1,...] where DELTA is the operator defined in A084938.
A094645*A007318 as infinite lower triangular matrices.
Row sums are the factorial numbers. - Roger L. Bagula, Apr 18 2008
Exponential Riordan array [1/(1-x), log(1/(1-x))]. - Ralf Stephan, Feb 07 2014
Also the Bell transform of the factorial numbers (A000142). For the definition of the Bell transform see A264428 and for cross-references A265606. - Peter Luschny, Dec 31 2015
This is the lower triagonal Sheffer matrix of the associated or Jabotinsky type |S1| = (1, -log(1-x)) (see the W. Lang link under A006232 for the notation and references). This implies the e.g.f.s given below. |S1| is the transition matrix from the monomial basis {x^n} to the rising factorial basis {risefac(x,n)}, n >= 0. - Wolfdieter Lang, Feb 21 2017
T(n, k), for n >= k >= 1, is also the total volume of the n-k dimensional cell (polytope) built from the n-k orthogonal vectors of pairwise different lengths chosen from the set {1, 2, ..., n-1}. See the elementary symmetric function formula for T(n, k) and an example below. - Wolfdieter Lang, May 28 2017
From Wolfdieter Lang, Jul 20 2017: (Start)
The compositional inverse w.r.t. x of y = y(t;x) = x*(1 - t(-log(1-x)/x)) = x + t*log(1-x) is x = x(t;y) = ED(y,t) := Sum_{d>=0} D(d,t)*y^(d+1)/(d+1)!, the e.g.f. of the o.g.f.s D(d,t) = Sum_{m>=0} T(d+m, m)*t^m of the diagonal sequences of the present triangle. See the P. Bala link for a proof (there d = n-1, n >= 1, is the label for the diagonals).
This inversion gives D(d,t) =  P(d, t)/(1-t)^(2*d+1), with the numerator polynomials P(d, t) =  Sum_{m=0..d} A288874(d, m)*t^m. See an example below. See also the P. Bala formula in A112007. (End)
For n > 0, T(n,k) is the number of permutations of the integers from 1 to n which have k visible digits when viewed from a specific end, in the sense that a higher value hides a lower one in a subsequent position. - Ian Duff, Jul 12 2019

			Triangle T(n,k) begins:
  1;
  0,    1;
  0,    1,     1;
  0,    2,     3,     1;
  0,    6,    11,     6,    1;
  0,   24,    50,    35,   10,    1;
  0,  120,   274,   225,   85,   15,   1;
  0,  720,  1764,  1624,  735,  175,  21,  1;
  0, 5040, 13068, 13132, 6769, 1960, 322, 28, 1;
  ...
---------------------------------------------------
Production matrix is
  0, 1
  0, 1, 1
  0, 1, 2,  1
  0, 1, 3,  3,  1
  0, 1, 4,  6,  4,  1
  0, 1, 5, 10, 10,  5,  1
  0, 1, 6, 15, 20, 15,  6, 1
  0, 1, 7, 21, 35, 35, 21, 7, 1
  ...
From _Wolfdieter Lang_, May 09 2017: (Start)
Three term recurrence: 50 = T(5, 2) = 1*6 + (5-1)*11 = 50.
Recurrence from the Sheffer a-sequence [1, 1/2, 1/6, 0, ...]: 50 = T(5, 2) = (5/2)*(binomial(1, 1)*1*6 + binomial(2, 1)*(1/2)*11 + binomial(3, 1)*(1/6)*6 + 0) = 50. The vanishing z-sequence produces the k=0 column from T(0, 0) = 1. (End)
Elementary symmetric function T(4, 2) = sigma^{(3)}_2 = 1*2 + 1*3 + 2*3 = 11. Here the cells (polytopes) are 3 rectangles with total area 11. - _Wolfdieter Lang_, May 28 2017
O.g.f.s of diagonals: d=2 (third diagonal) [0, 6, 50, ...] has D(2,t) = P(2, t)/(1-t)^5, with P(2, t) = 2 + t, the n = 2 row of A288874. - _Wolfdieter Lang_, Jul 20 2017
Boas-Buck recurrence for column k = 2 and n = 5: T(5, 2) = (5!*2/3)*((3/8)*T(2,2)/2! + (5/12)*T(3,2)/3! + (1/2)*T(4,2)/4!) = (5!*2/3)*(3/16 + (5/12)*3/3! + (1/2)*11/4!) = 50. The beta sequence begins: {1/2, 5/12, 3/8, ...}. - _Wolfdieter Lang_, Aug 11 2017

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 31, 187, 441, 996.
R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., Table 259, p. 259.
Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 149-150

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Roland Bacher and P. De La Harpe, Conjugacy growth series of some infinitely generated groups, hal-01285685v2, 2016.
Eli Bagno and David Garber, Combinatorics of q,r-analogues of Stirling numbers of type B, arXiv:2401.08365 [math.CO], 2024. See page 5.
Peter Bala, Diagonals of triangles with generating function exp(t*F(x)).
J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure, arXiv:1307.2010 [math.CO], 2013.
Jean-Luc Baril and Sergey Kirgizov, The pure descent statistic on permutations, Preprint, 2016.
Jean-Luc Baril and Sergey Kirgizov, Transformation à la Foata for special kinds of descents and excedances, arXiv:2101.01928 [math.CO], 2021.
Jean-Luc Baril and José L. Ramírez, Some distributions in increasing and flattened permutations, arXiv:2410.15434 [math.CO], 2024. See p. 9.
Ricky X. F. Chen, A Note on the Generating Function for the Stirling Numbers of the First Kind, Journal of Integer Sequences, 18 (2015), #15.3.8.
Bishal Deb and Alan D. Sokal, Higher-order Stirling cycle and subset triangles: Total positivity, continued fractions and real-rootedness, arXiv:2507.18959 [math.CO], 2025. See p. 5.
FindStat - Combinatorial Statistic Finder, The number of saliances of a permutation, The number of cycles in the cycle decomposition of a permutation.
Bill Gosper, Colored illustrations of triangle of Stirling numbers of first kind read mod 2, 3, 4, 5, 6, 7
W. Steven Gray and Makhin Thitsa, System Interconnections and Combinatorial Integer Sequences, in: System Theory (SSST), 2013 45th Southeastern Symposium on, Date of Conference: 11-11 March 2013, Digital Object Identifier: 10.1109/SSST.2013.6524939.
John M. Holte, Carries, Combinatorics and an Amazing Matrix, The American Mathematical Monthly, Vol. 104, No. 2 (Feb., 1997), pp. 138-149.
Tanya Khovanova and J. B. Lewis, Skyscraper Numbers, J. Int. Seq. 16 (2013) #13.7.2.
Sergey Kitaev and Philip B. Zhang, Distributions of mesh patterns of short lengths, arXiv:1811.07679 [math.CO], 2018.
Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers, arXiv:1707.04451 [math.NT], 2017.
Shuzhen Lv and Philip B. Zhang, Joint equidistributions of mesh patterns 123 and 321 with symmetric and antipodal shadings, arXiv:2501.00357 [math.CO], 2024. See p. 11.
Shuzhen Lv and Philip B. Zhang, Joint equidistributions of mesh patterns 123 and 132 with antipodal shadings, arXiv:2506.23148 [math.CO], 2025. See p. 6.
Shi-Mei Ma, Some combinatorial sequences associated with context-free grammars, arXiv:1208.3104v2 [math.CO], 2012. - From _N. J. A. Sloane_, Aug 21 2012
Mathematics Stack Exchange, Symmetric (under the swapping) recursions for Stirling numbers of both kinds, Aug 10 2025.
Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.
Emanuele Munarini, Combinatorial identities involving the central coefficients of a Sheffer matrix, Applicable Analysis and Discrete Mathematics (2019) Vol. 13, 495-517.
Fedor Petrov, Recursive algorithm for columns of Stirling numbers of the first kind, answer to question on MathOverflow (2025).
E. G. Santos, Counting non-attacking chess pieces placements: Bishops and Anassas, arXiv:2411.16492 [math.CO], 2024. See p. 2.
Umesh Shankar, Log-concavity of rows of triangular arrays satisfying a certain super-recurrence, arXiv:2508.12467 [math.CO], 2025. See p. 4.
Xun-Tuan Su, Deng-Yun Yang, and Wei-Wei Zhang, A note on the generalized factorial, Australasian Journal of Combinatorics, Volume 56 (2013), Pages 133-137.
Benjamin Testart, Completing the enumeration of inversion sequences avoiding one or two patterns of length 3, arXiv:2407.07701 [math.CO], 2024. See p. 37.

Essentially a duplicate of A048994. Cf. A008275, A008277, A112007, A130534, A288874, A354795.

a132393 n k = a132393_tabl !! n !! k
a132393_row n = a132393_tabl !! n
a132393_tabl = map (map abs) a048994_tabl
-- Reinhard Zumkeller, Nov 06 2013

a132393_row := proc(n) local k; seq(coeff(expand(pochhammer (x,n)),x,k),k=0..n) end: # Peter Luschny, Nov 28 2010

p[t_] = 1/(1 - t)^x; Table[ ExpandAll[(n!)SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Table[(n!)* CoefficientList[SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}]; Flatten[a] (* Roger L. Bagula, Apr 18 2008 *)
Flatten[Table[Abs[StirlingS1[n,i]],{n,0,10},{i,0,n}]] (* Harvey P. Dale, Feb 04 2014 *)

create_list(abs(stirling1(n,k)),n,0,12,k,0,n); /* Emanuele Munarini, Mar 11 2011 */

column(n,k) = my(v1, v2); v1 = vector(n-1, i, 0); v2 = vector(n, i, 0); v2[1] = 1; for(i=1, n-1, v1[i] = (i+k)*(i+k-1)/2*v2[i]; for(j=1, i-1, v1[j] *= (i-j)*(i+k)/(i-j+2)); v2[i+1] = vecsum(v1)/i); v2 \\ generates n first elements of the k-th column starting from the first nonzero element. - Mikhail Kurkov, Mar 05 2025

T(n,k) = T(n-1,k-1)+(n-1)*T(n-1,k), n,k>=1; T(n,0)=T(0,k); T(0,0)=1.
Sum_{k=0..n} T(n,k)*x^(n-k) = A000012(n), A000142(n), A001147(n), A007559(n), A007696(n), A008548(n), A008542(n), A045754(n), A045755(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively. - Philippe Deléham, Nov 13 2007
Expand 1/(1-t)^x = Sum_{n>=0}p(x,n)*t^n/n!; then the coefficients of the p(x,n) produce the triangle. - Roger L. Bagula, Apr 18 2008
Sum_{k=0..n} T(n,k)*2^k*x^(n-k) = A000142(n+1), A000165(n), A008544(n), A001813(n), A047055(n), A047657(n), A084947(n), A084948(n), A084949(n) for x = 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively. - Philippe Deléham, Sep 18 2008
a(n) = Sum_{k=0..n} T(n,k)*3^k*x^(n-k) = A001710(n+2), A001147(n+1), A032031(n), A008545(n), A047056(n), A011781(n), A144739(n), A144756(n), A144758(n) for x=1,2,3,4,5,6,7,8,9,respectively. - Philippe Deléham, Sep 20 2008
Sum_{k=0..n} T(n,k)*4^k*x^(n-k) = A001715(n+3), A002866(n+1), A007559(n+1), A047053(n), A008546(n), A049308(n), A144827(n), A144828(n), A144829(n) for x=1,2,3,4,5,6,7,8,9 respectively. - Philippe Deléham, Sep 21 2008
Sum_{k=0..n} x^k*T(n,k) = x*(1+x)*(2+x)*...*(n-1+x), n>=1. - Philippe Deléham, Oct 17 2008
From Wolfdieter Lang, Feb 21 2017: (Start)
E.g.f. k-th column: (-log(1 - x))^k, k >= 0.
E.g.f. triangle (see the Apr 18 2008 Baluga comment): exp(-x*log(1-z)).
E.g.f. a-sequence: x/(1 - exp(-x)). See A164555/A027642. The e.g.f. for the z-sequence is 0. (End)
From Wolfdieter Lang, May 28 2017: (Start)
The row polynomials R(n, x) = Sum_{k=0..n} T(n, k)*x^k, for n >= 0, are R(n, x) = risefac(x,n-1) := Product_{j=0..n-1} x+j, with the empty product for n=0 put to 1. See the Feb 21 2017 comment above. This implies:
T(n, k) = sigma^{(n-1)}_(n-k), for n >= k >= 1, with the elementary symmetric functions sigma^{(n-1)}_m of degree m in the n-1 symbols 1, 2, ..., n-1, with binomial(n-1, m) terms. See an example below.(End)
Boas-Buck type recurrence for column sequence k: T(n, k) = (n!*k/(n - k)) * Sum_{p=k..n-1} beta(n-1-p)*T(p, k)/p!, for n > k >= 0, with input T(k, k) = 1, and beta(k) = A002208(k+1)/A002209(k+1). See a comment and references in A286718. - Wolfdieter Lang, Aug 11 2017
T(n,k) = Sum_{j=k..n} j^(j-k)*binomial(j-1, k-1)*A354795(n,j) for n > 0. - Mélika Tebni, Mar 02 2023
n-th row polynomial: n!*Sum_{k = 0..2*n} (-1)^k*binomial(-x, k)*binomial(-x, 2*n-k) = n!*Sum_{k = 0..2*n} (-1)^k*binomial(1-x, k)*binomial(-x, 2*n-k). - Peter Bala, Mar 31 2024
From Mikhail Kurkov, Mar 05 2025: (Start)
For a general proof of the formulas below via generating functions, see Mathematics Stack Exchange link.
Recursion for the n-th row (independently of other rows): T(n,k) = 1/(n-k)*Sum_{j=2..n-k+1} binomial(-k,j)*T(n,k+j-1)*(-1)^j for 1 <= k < n with T(n,n) = 1.
Recursion for the k-th column (independently of other columns): T(n,k) = 1/(n-k)*Sum_{j=2..n-k+1} (j-2)!*binomial(n,j)*T(n-j+1,k) for 1 <= k < n with T(n,n) = 1 (see Fedor Petrov link). (End)

A052562 a(n) = 5^n * n!.

Original entry on oeis.org

1, 5, 50, 750, 15000, 375000, 11250000, 393750000, 15750000000, 708750000000, 35437500000000, 1949062500000000, 116943750000000000, 7601343750000000000, 532094062500000000000, 39907054687500000000000

Offset: 0

Joe Keane (jgk(AT)jgk.org)

A simple regular expression in a labeled universe.

For n >= 1 a(n) is the order of the wreath product of the symmetric group S_n and the Abelian group (C_5)^n. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 07 2001

Vincenzo Librandi, Table of n, a(n) for n = 0..300
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 504
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

Cf. A000142, A008548, A008546, A034325, A000165, A047055, A047056, A051150, A092514, A092618.

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

spec := [S,{S=Sequence(Union(Z,Z,Z,Z,Z))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
with(combstruct):A:=[N,{N=Cycle(Union(Z$5))},labeled]: seq(count(A,size=n)/5,n=1..16); # Zerinvary Lajos, Dec 05 2007

Table[5^n*n!, {n, 0, 20}] (* Wesley Ivan Hurt, Sep 28 2013 *)

{a(n) = 5^n*n!}; \\ G. C. Greubel, May 05 2019

[5^n*factorial(n) for n in (0..20)] # G. C. Greubel, May 05 2019

a(n) = A051150(n+1, 0) (first column of triangle).
E.g.f.: 1/(1-5*x).
a(n) = 5*n*a(n-1) with a(0)=1.
G.f.: 1/(1-5*x/(1-5*x/(1-10*x/(1-10*x/(1-15*x/(1-15*x/(1-20*x/(1-... (continued fraction). - Philippe Deléham, Jan 08 2012
G.f.: 1/Q(0), where Q(k) = 1 - 5*x*(2*k+1) - 25*x^2*(k+1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Sep 28 2013
a(n) = n!*A000351(n). - R. J. Mathar, Aug 21 2014
From Amiram Eldar, Jun 25 2020: (Start)
Sum_{n>=0} 1/a(n) = e^(1/5) (A092514).
Sum_{n>=0} (-1)^n/a(n) = e^(-1/5) (A092618). (End)

Name changed by Arkadiusz Wesolowski, Oct 04 2011

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)

A085157 Quintuple factorials, 5-factorials, n!!!!!, n!5.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 14, 24, 36, 50, 66, 168, 312, 504, 750, 1056, 2856, 5616, 9576, 15000, 22176, 62832, 129168, 229824, 375000, 576576, 1696464, 3616704, 6664896, 11250000, 17873856, 54286848, 119351232, 226606464, 393750000, 643458816

Offset: 0

Hugo Pfoertner, Jun 21 2003

nonn

The term "Quintuple factorial numbers" is also used for the sequences A008546, A008548, A052562, A047055, A047056 which have a different definition. The definition given here is the one commonly used.

			a(12) = 168 because 12*a(12-5) = 12*a(7) = 12*14 = 168.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Peter Luschny, Multifactorials
Eric Weisstein's World of Mathematics, Multifactorial.

Cf. n!:A000142, n!!:A006882, n!!!:A007661, n!!!!:A007662, n!!!!!!:A085158, 5-factorial primes: n!!!!!+1:A085148, n!!!!!-1:A085149.

Cf. A288092.

a:= function(n)
    if n<1 then return 1;
    else return n*a(n-5);
    fi;
  end;
List([0..40], n-> a(n) ); # G. C. Greubel, Aug 18 2019

b:= func< n | (n lt 6) select n else n*Self(n-5) >;
[1] cat [b(n): n in [1..40]]; // G. C. Greubel, Aug 18 2019

a:= n-> `if`(n < 1, 1, n*a(n-5)) end proc; seq(a(n), n = 0..40); # G. C. Greubel, Aug 18 2019

a[n_]:= If[n<1, 1, n*a[n-5]]; Table[a[n], {n,0,40}] (* G. C. Greubel, Aug 18 2019 *)
Table[Times@@Range[n,1,-5],{n,0,40}] (* Harvey P. Dale, May 12 2020 *)

a(n)=if(n<1, 1, n*a(n-5))
for(n=0,50,print1(a(n),",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 19 2006

def A085157(n):
    if n <= 0:
        return 1
    else:
        return n*A085157(n-5)
n = 0
while n <= 40:
    print(n,A085157(n))
    n = n+1 # A.H.M. Smeets, Aug 18 2019

def a(n):
    if (n<1): return 1
    else: return n*a(n-5)
[a(n) for n in (0..40)] # G. C. Greubel, Aug 18 2019

a(n) = 1 for n < 1, otherwise a(n) = n*a(n-5).

Sum_{n>=0} 1/a(n) = A288092. - Amiram Eldar, Nov 10 2020

A049209 a(n) = -Product_{k=0..n} (7*k-1); sept-factorial numbers.

Original entry on oeis.org

1, 6, 78, 1560, 42120, 1432080, 58715280, 2818333440, 155008339200, 9610517030400, 663125675097600, 50397551307417600, 4182996758515660800, 376469708266409472000, 36517561701841718784000, 3797826416991538753536000, 421558732286060801642496000

Offset: 0

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 0..330
Index entries for sequences related to factorial numbers.

Cf. A034833, A045754, A048994, A051188, A084947, A144739, A144827, A147585.
Row sums of triangle A051186 (scaled Stirling1 triangle).
Sequences of the form m^n*Pochhammer((m-1)/m, n): A000007 (m=1), A001147 (m=2), A008544 (m=3), A008545 (m=4), A008546 (m=5), A008543 (m=6), this sequence (m=7), A049210 (m=8), A049211 (m=9), A049212 (m=10), A254322 (m=11), A346896 (m=12).

[ -&*[ (7*k-1): k in [0..n-1] ]: n in [1..15] ]; // Klaus Brockhaus, Nov 10 2008

CoefficientList[Series[(1-7*x)^(-6/7),{x,0,20}],x] * Range[0,20]! (* Vaclav Kotesovec, Jan 28 2015 *)
With[{m=7}, Table[m^n*Pochhammer[(m-1)/m, n], {n, 0, 30}]] (* G. C. Greubel, Feb 16 2022 *)

m=7; [m^n*rising_factorial((m-1)/m, n) for n in (0..30)] # G. C. Greubel, Feb 16 2022

a(n) = 6*A034833(n) = (7*n-1)*(!^7), n >= 1, a(0) := 1.
a(n) = Product_{k=1..n} (7*k - 1). a(0) = 1; a(n) = (7*n - 1)*a(n-1) for n > 0. - Klaus Brockhaus, Nov 10 2008
G.f.: 1/(1-6*x/(1-7*x/(1-13*x/(1-14*x/(1-20*x/(1-21*x/(1-27*x/(1-28*x/(1-...(continued fraction). - Philippe Deléham, Jan 08 2012
a(n) = (-1)^n*Sum_{k=0..n} 7^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) = 7^n * Gamma(n+6/7) / Gamma(6/7). - Vaclav Kotesovec, Jan 28 2015
E.g.f.: (1-7*x)^(-6/7). - Vaclav Kotesovec, Jan 28 2015
From Nikolaos Pantelidis, Dec 19 2020: (Start)
G.f.: 1/G(0) where G(k) = 1 - (14*k+6)*x - 7*(k+1)*(7*k+6)*x^2/G(k+1); (continued fraction).
which starts as 1/(1-6*x-42*x^2/(1-20*x-182*x^2/(1-34*x-420*x^2/(1-48*x-756*x^2/(1-62*x-1190*x^2/(1-... )))))) (Jacobi continued fraction).
G.f.: 1/Q(0) where Q(k) = 1 - (7*k+6)*x/(1 - (7*k+7)*x/Q(k+1) ); (continued fraction). (End)
Sum_{n>=0} 1/a(n) = 1 + (e/7)^(1/7)*(Gamma(6/7) - Gamma(6/7, 1/7)). - Amiram Eldar, Dec 19 2022

A034301 a(n) = n-th quintic factorial number divided by 4.

Original entry on oeis.org

1, 9, 126, 2394, 57456, 1666224, 56651616, 2209413024, 97214173056, 4763494479744, 257228701906176, 15176493412464384, 971295578397720576, 67019394909442719744, 4959435223298761261056, 391795382640602139623424, 32910812141810579728367616, 2929062280621141595824717824

Offset: 1

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 1..350

Cf. A008546, A008548, A034300, A025750.

List([1..20], n-> Product([1..n], k-> 5*k-1)/4 ); # G. C. Greubel, Aug 23 2019

[&*[5*k-1: k in [1..n]]/4: n in [1..20]]; // G. C. Greubel, Aug 23 2019

a:= n-> mul(5*k-1, k=1..n)/4: seq(a(n), n=1..20); # G. C. Greubel, Aug 23 2019

Table[-5^(n+1)*Pochhammer[-1/5, n+1]/4, {n,20}] (* G. C. Greubel, Aug 23 2019 *)

a(n) = prod(k=1,n, 5*k-1)/4;
vector(20, n, a(n)) \\ G. C. Greubel, Aug 23 2019

[-5^(n+1)*rising_factorial(-1/5, n+1)/4 for n in (1..20)] # G. C. Greubel, Aug 23 2019

a(n) = A008546(n)/4.
4*a(n) = (5*n-1)(!^5) = Product_{j=1..n} (5*j-1).
a(n) = (5*n)!/(5^n*n!*A008548(n)*2*A034323(n)*3*A034300(n)).
E.g.f.: (-1 + (1-5*x)^(-4/5))/4, a(0) = 0.
a(n) ~ sqrt(2*Pi) * 5/(4*Gamma(4/5)) * n^(13/10) * (5*n/e)^n * (1 + (241/300)/n + ...). - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
D-finite with recurrence: a(n) +(-5*n+1)*a(n-1)=0. - R. J. Mathar, Feb 20 2020
Sum_{n>=1} 1/a(n) = 4*(e/5)^(1/5)*(Gamma(4/5) - Gamma(4/5, 1/5)). - Amiram Eldar, Dec 19 2022

Terms a(17) onward added by G. C. Greubel, Aug 23 2019

A047056 Quintuple factorial numbers: Product_{k=0..n-1} (5*k+3).

Original entry on oeis.org

1, 3, 24, 312, 5616, 129168, 3616704, 119351232, 4535346816, 195019913088, 9360955828224, 496130658895872, 28775578215960576, 1812861427605516288, 123274577077175107584, 8999044126633782853632, 701925441877435062583296, 58259811675827110194413568

Offset: 0

Joe Keane (jgk(AT)jgk.org)

G. C. Greubel, Table of n, a(n) for n = 0..350

Cf. A008546, A008548, A047055, A048994, A052562.

List([0..20], n-> Product([0..n-1], k-> 5*k+3 )); # G. C. Greubel, Aug 20 2019

[1] cat [(&*[5*k+3: k in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Aug 20 2019

seq(mul(5*k+3, k = 0..n-1), n = 0..20); # G. C. Greubel, Aug 20 2019

Table[5^n*Pochhammer[3/5, n], {n,0,20}] (* G. C. Greubel, Aug 20 2019 *)
Join[{1},FoldList[Times,5*Range[0,20]+3]] (* Harvey P. Dale, Oct 08 2020 *)

vector(20, n, n--; prod(j=0,n-1, 5*j+3) ) \\ G. C. Greubel, Aug 20 2019

[5^n*rising_factorial(3/5, n) for n in (0..20)] # G. C. Greubel, Aug 20 2019

E.g.f.: (1-5*x)^(-3/5).
a(n) ~ sqrt(2*Pi)/Gamma(3/5)*n^(1/10)*(5*n/e)^n*(1 - 11/300*n^ - 1 + ...). - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
G.f.: 1/(1-3*x/(1-5*x/(1-8*x/(1-10*x/(1-13*x/(1-15*x/(1-18*x/(1-20*x/(1- ... (continued fraction). - Philippe Deléham, Jan 08 2012
a(n) = (-2)^n*Sum_{k=0..n} (5/2)^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
G.f.: 1/G(0)  where G(k) = 1 - x*(5*k+3)/( 1 - 5*x*(k+1)/G(k+1) ); (continued fraction ). - Sergei N. Gladkovskii, Mar 23 2013
G.f.: G(0)/2, where G(k)= 1  + 1/(1 - (5*k+3)*x/((5*k+3)*x + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 14 2013
D-finite with recurrence: a(n) +(-5*n+2)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
Sum_{n>=0} 1/a(n) = 1 + (e/5^2)^(1/5)*(Gamma(3/5) - Gamma(3/5, 1/5)). - Amiram Eldar, Dec 19 2022

A049211 a(n) = Product_{k=1..n} (9*k - 1); 9-factorial numbers.

Original entry on oeis.org

1, 8, 136, 3536, 123760, 5445440, 288608320, 17893715840, 1270453824640, 101636305971200, 9045631231436800, 886471860680806400, 94852489092846284800, 11002888734770169036800, 1375361091846271129600000, 184298386307400331366400000, 26354669241958247385395200000

Offset: 0

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 0..325

Sequences of the form m^n*Pochhammer((m-1)/m, n): A000007 (m=1), A001147 (m=2), A008544 (m=3), A008545 (m=4), A008546 (m=5), A008543 (m=6), A049209 (m=7), A049210 (m=8), this sequence (m=9), A049212 (m=10), A254322 (m=11), A346896 (m=12).

Cf. A035022, A048994.

m:=9; [Round(m^n*Gamma(n +(m-1)/m)/Gamma((m-1)/m)): n in [0..20]]; // G. C. Greubel, Feb 08 2022

CoefficientList[Series[(1-9*x)^(-8/9),{x,0,20}],x] * Range[0,20]! (* Vaclav Kotesovec, Jan 28 2015 *)

a(n) = prod(k=1, n, 9*k-1); \\ Michel Marcus, Jan 08 2015

m=9; [m^n*rising_factorial((m-1)/m, n) for n in (0..20)] # G. C. Greubel, Feb 08 2022

a(n) = 8*A035022(n) = (9*n-1)(!^9), n >= 1, a(0) = 1.
a(n) = (-1)^n*Sum_{k=0..n} 9^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) = 9^n * Gamma(n+8/9) / Gamma(8/9). - Vaclav Kotesovec, Jan 28 2015
E.g.f: (1-9*x)^(-8/9). - Vaclav Kotesovec, Jan 28 2015
From Nikolaos Pantelidis, Dec 09 2020: (Start)
G.f.: 1/(1-8*x-72*x^2/(1-26*x-306*x^2/(1-44*x-702*x^2/(1-62*x-1260*x^2/(1-80*x-1980*x^2/(1-...)))))) (Jacobi continued fraction).
G.f.: 1/(1-8*x/(1-9*x/(1-17*x/(1-18*x/(1-26*x/(1-27*x/(1-35*x/(1-36*x/(1-44*x/(1-45*x/(1-...))))))))))) (Stieltjes continued fraction). (End)
From Nikolaos Pantelidis, Dec 19 2020: (Start)
G.f.: 1/G(0) where G(k) = 1 - (18*k+8)*x - 9*(k+1)*(9*k+8)*x^2/G(k+1) (continued fraction).
G.f.: 1/Q(0) where Q(k) = 1 - x*(9*k+8)/(1 - x*(9*k+9)/Q(k+1) ) (continued fraction). (End)
G.f.: hypergeometric2F0([1, 8/9], [--], 9*x). - G. C. Greubel, Feb 08 2022
Sum_{n>=0} 1/a(n) = 1 + (e/9)^(1/9)*(Gamma(8/9) - Gamma(8/9, 1/9)). - Amiram Eldar, Dec 21 2022

a(9) (originally given incorrectly as 1011636305971200) corrected by Peter Bala, Feb 20 2015
a(15)-a(16) from Vincenzo Librandi, Feb 20 2015
a(16) corrected and incorrect MAGMA program removed by Georg Fischer, May 10 2021

cp's OEIS Frontend

A112940 INVERT transform (with offset) of quintuple factorials (A008546), where g.f. satisfies: A(x) = 1 + x*[d/dx x*A(x)^5]/A(x)^5.

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A112941 Logarithmic derivative of A112940 such that a(n)=(1/5)*A112940(n+1) for n>0, where A112940 equals the INVERT transform (with offset) of quintuple factorials A008546.

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A132393 Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.

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A052562 a(n) = 5^n * 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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A085157 Quintuple factorials, 5-factorials, n!!!!!, n!5.

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A112940 INVERT transform (with offset) of quintuple factorials (A008546), where g.f. satisfies: A(x) = 1 + x[d/dx xA(x)^5]/A(x)^5.

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.