A144828 -id:A144828 - cp's OEIS frontend

A001813 Quadruple factorial numbers: a(n) = (2n)!/n!.

1, 2, 12, 120, 1680, 30240, 665280, 17297280, 518918400, 17643225600, 670442572800, 28158588057600, 1295295050649600, 64764752532480000, 3497296636753920000, 202843204931727360000, 12576278705767096320000, 830034394580628357120000, 58102407620643984998400000

Offset: 0

N. J. A. Sloane

Counts binary rooted trees (with out-degree <= 2), embedded in plane, with n labeled end nodes of degree 1. Unlabeled version gives Catalan numbers A000108.
Define a "downgrade" to be the permutation which places the items of a permutation in descending order. We are concerned with permutations that are identical to their downgrades. Only permutations of order 4n and 4n+1 can have this property; the number of permutations of length 4n having this property are equinumerous with those of length 4n+1. If a permutation p has this property then the reversal of this permutation also has it. a(n) = number of permutations of length 4n and 4n+1 that are identical to their downgrades. - Eugene McDonnell (eemcd(AT)mac.com), Oct 26 2003
Number of broadcast schemes in the complete graph on n+1 vertices, K_{n+1}. - Calin D. Morosan (cd_moros(AT)alumni.concordia.ca), Nov 28 2008
Hankel transform is A137565. - Paul Barry, Nov 25 2009
The e.g.f. of 1/a(n) = n!/(2*n)! is (exp(sqrt(x)) + exp(-sqrt(x)) )/2. - Wolfdieter Lang, Jan 09 2012
From Tom Copeland, Nov 15 2014: (Start)
Aerated with intervening zeros (1,0,2,0,12,0,120,...) = a(n) (cf. A123023 and A001147), the e.g.f. is e^(t^2), so this is the base for the Appell sequence with e.g.f. e^(t^2) e^(x*t) = exp(P(.,x),t) (reverse A059344, cf. A099174, A066325 also). P(n,x) = (a. + x)^n with (a.)^n = a_n and comprise the umbral compositional inverses for e^(-t^2)e^(x*t) = exp(UP(.,x),t), i.e., UP(n,P(.,t)) = x^n = P(n,UP(.,t)), e.g., (P(.,t))^n = P(n,t).
Equals A000407*2 with leading 1 added. (End)
a(n) is also the number of square roots of any permutation in S_{4*n} whose disjoint cycle decomposition consists of 2*n transpositions. - Luis Manuel Rivera Martínez, Mar 04 2015
Self-convolution gives A076729. - Vladimir Reshetnikov, Oct 11 2016
For n > 1, it follows from the formula dated Aug 07 2013 that a(n) is a Zumkeller number (A083207). - Ivan N. Ianakiev, Feb 28 2017
For n divisible by 4, a(n/4) is the number of ways to place n points on an n X n grid with pairwise distinct abscissae, pairwise distinct ordinates, and 90-degree rotational symmetry. For n == 1 (mod 4), the number of ways is a((n-1)/4) because the center point can be considered "fixed". For 180-degree rotational symmetry see A006882, for mirror symmetry see A000085, A135401, and A297708. - Manfred Scheucher, Dec 29 2017

			The following permutations of order 8 and their reversals have this property:
  1 7 3 5 2 4 0 6
  1 7 4 2 5 3 0 6
  2 3 7 6 1 0 4 5
  2 4 7 1 6 0 3 5
  3 2 6 7 0 1 5 4
  3 5 1 7 0 6 2 4

D. E. Knuth, The Art of Computer Programming, Vol. 4, Section 7.2.1.6, Eq. 32.
L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.
Eugene McDonnell, "Magic Squares and Permutations" APL Quote-Quad 7.3 (Fall, 1976)
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 0..100
Marcello Artioli, Giuseppe Dattoli, Silvia Licciardi, and Rosa Maria Pidatella, Hermite and Laguerre Functions: a Unifying Point of View, Università degli Studi di Catania, Sicily, Italy (2019).
Murray Bremner and Martin Markl, Distributive laws between the Three Graces, arXiv:1809.08191 [math.AT], 2018.
R. B. Brent, Generalizing Tuenter's Binomial Sums, J. Int. Seq. 18 (2015) # 15.3.2.
Peter J. Cameron, Some treelike objects, Quart. J. Math. Oxford Ser. (2) 38 (1987), no. 150, 155--183. MR0891613 (89a:05009). See p. 155. - _N. J. A. Sloane_, Apr 18 2014
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Elliot J. Carr and Matthew J. Simpson, New homogenization approaches for stochastic transport through heterogeneous media, arXiv:1810.08890 [physics.bio-ph], 2018.
W. Y. C. Chen, L. W. Shapiro and L. L. M. Young, Parity reversing involutions on plane trees and 2-Motzkin paths, arXiv:math/0503300 [math.CO], 2005.
Ali Chouria, Vlad-Florin Drǎgoi, and Jean-Gabriel Luque, On recursively defined combinatorial classes and labelled trees, arXiv:2004.04203 [math.CO], 2020.
P. Cvitanovic, Group theory for Feynman diagrams in non-Abelian gauge theories, Phys. Rev. D14 (1976), 1536-1553.
Nick Early, Honeycomb tessellations and canonical bases for permutohedral blades, arXiv:1810.03246 [math.CO], 2018.
John Engbers, David Galvin, and Clifford Smyth, Restricted Stirling and Lah numbers and their inverses, arXiv:1610.05803 [math.CO], 2016. See p. 8.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 127
S. Goodenough and C. Lavault, On subsets of Riordan subgroups and Heisenberg--Weyl algebra, arXiv preprint arXiv:1404.1894 [cs.DM], 2014.
S. Goodenough and C. Lavault, Overview on Heisenberg—Weyl Algebra and Subsets of Riordan Subgroups, The Electronic Journal of Combinatorics, 22(4) (2015), #P4.16,
H. W. Gould, A class of binomial sums and a series transform, Utilitas Math., 45 (1994), 71-83. (Annotated scanned copy)
A. M. Ibrahim, Extension of factorial concept to negative numbers, Notes on Number Theory and Discrete Mathematics, Vol. 19, 2013, 2, 30-42.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 115
L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181. [Annotated scan of pages 180 and 181 only]
Jesús Leaños, Rutilo Moreno, and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, Australas. J. Combin. 52 (2012), 41-54 (Theorem 1).
Jesús Leaños, Rutilo Moreno, and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, arXiv:1005.1531 [math.CO], 2010-2011.
Édouard Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1, p. 221.
R. J. Marsh and P. P. Martin, Pascal arrays: counting Catalan sets, arXiv:math/0612572 [math.CO], 2006.
Calin D. Morosan, On the number of broadcast schemes in networks, Information Processing Letters, Volume 100, Issue 5 (2006), 188-193.
R. A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
C. Radoux, Déterminants de Hankel et théorème de Sylvester, Séminaire Lotharingien de Combinatoire, B28b (1992), 9 pp.
J. Riordan, Letter to N. J. A. Sloane, Feb 03 1975 (with notes by njas)
H. E. Salzer, Coefficients for expressing the first thirty powers in terms of the Hermite polynomials, Math. Comp., 3 (1948), 167-169.
Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See pp. 12-13.
Index to divisibility sequences
Index entries for related partition-counting sequences

Cf. A037224, A048854, A001147, A007696, A008545, A122670 (essentially the same sequence), A000165, A047055, A047657, A084947, A084948, A084949, A010050, A000142, A008275, A000108, A000984, A008276, A000680, A094216.
Catalan(n-1)*M^(n-1)*n! for M=1,2,3,4,5,6: A001813, A052714 (or A144828), A221954, A052734, A221953, A221955.
Cf. A123023, A059344, A099174, A066325, A001700, A000407, A006882.
Cf. A000085, A135401, A297708. - Manfred Scheucher, Jan 07 2018

List([0..20],n->Factorial(2*n)/Factorial(n)); # Muniru A Asiru, Nov 01 2018

[Factorial(2*n)/Factorial(n): n in [0..20]]; // Vincenzo Librandi, Oct 09 2018

A001813 := n->(2*n)!/n!;
A001813 := n -> mul(k, k = select(k-> k mod 4 = 2,[$1 .. 4*n])):
seq(A001813(n), n=0..16);  # Peter Luschny, Jun 23 2011

Table[(2n)!/n!, {n,0,20}] (* Harvey P. Dale, May 02 2011 *)

makelist(binomial(n+n, n)*n!,n,0,30); /* Martin Ettl, Nov 05 2012 */

a(n)=binomial(n+n,n)*n! \\ Charles R Greathouse IV, Jun 15 2011

first(n) = x='x+O('x^n); Vec(serlaplace((1 - 4*x)^(-1/2))) \\ Iain Fox, Jan 01 2018 (corrected by Iain Fox, Jan 11 2018)

from math import factorial
def A001813(n): return factorial(n<<1)//factorial(n) # Chai Wah Wu, Feb 14 2023

[binomial(2*n,n)*factorial(n) for n in range(0, 17)] # Zerinvary Lajos, Dec 03 2009

E.g.f.: (1-4*x)^(-1/2).
a(n) = (2*n)!/n! = Product_{k=0..n-1} (4*k + 2) = A081125(2*n).
Integral representation as n-th moment of a positive function on a positive half-axis: a(n) = Integral_{x=0..oo} x^n*exp(-x/4)/(sqrt(x)*2*sqrt(Pi)) dx, n >= 0. This representation is unique. - Karol A. Penson, Sep 18 2001
Define a'(1)=1, a'(n) = Sum_{k=1..n-1} a'(n-k)*a'(k)*C(n, k); then a(n)=a'(n+1). - Benoit Cloitre, Apr 27 2003
With interpolated zeros (1, 0, 2, 0, 12, ...) this has e.g.f. exp(x^2). - Paul Barry, May 09 2003
a(n) = A000680(n)/A000142(n)*A000079(n) = Product_{i=0..n-1} (4*i + 2) = 4^n*Pochhammer(1/2, n) = 4^n*GAMMA(n+1/2)/sqrt(Pi). - Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003
For asymptotics, see the Robinson paper.
a(k) = (2*k)!/k! = Sum_{i=1..k+1} |A008275(i,k+1)| * k^(i-1). - André F. Labossière, Jun 21 2007
a(n) = 12*A051618(a) n >= 2. - Zerinvary Lajos, Feb 15 2008
a(n) = A000984(n)*A000142(n). - Zerinvary Lajos, Mar 25 2008
a(n) = A016825(n-1)*a(n-1). - Roger L. Bagula, Sep 17 2008
a(n) = (-1)^n*A097388(n). - D. Morosan (cd_moros(AT)alumni.concordia.ca), Nov 28 2008
From Paul Barry, Jan 15 2009: (Start)
G.f.: 1/(1-2x/(1-4x/(1-6x/(1-8x/(1-10x/(1-... (continued fraction);
a(n) = (n+1)!*A000108(n). (End)
a(n) = Sum_{k=0..n} A132393(n,k)*2^(2n-k). - Philippe Deléham, Feb 10 2009
G.f.: 1/(1-2x-8x^2/(1-10x-48x^2/(1-18x-120x^2/(1-26x-224x^2/(1-34x-360x^2/(1-42x-528x^2/(1-... (continued fraction). - Paul Barry, Nov 25 2009
a(n) = A173333(2*n,n) for n>0; cf. A006963, A001761. - Reinhard Zumkeller, Feb 19 2010
From Gary W. Adamson, Jul 19 2011: (Start)
a(n) = upper left term of M^n, M = an infinite square production matrix as follows:
  2, 2, 0, 0, 0, 0, ...
  4, 4, 4, 0, 0, 0, ...
  6, 6, 6, 6, 0, 0, ...
  8, 8, 8, 8, 8, 0, ...
  ...
(End)
a(n) = (-2)^n*Sum_{k=0..n} 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/Q(0), where Q(k) = 1 + x*(4*k+2) - x*(4*k+4)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 18 2013
G.f.: 2/G(0), where G(k) = 1 + 1/(1 - x*(8*k+4)/(x*(8*k+4) - 1 + 8*x*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 30 2013
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - 2*x/(2*x + 1/(2*k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 01 2013
D-finite with recurrence: a(n) = (4*n-6)*a(n-2) + (4*n-3)*a(n-1), n>=2. - Ivan N. Ianakiev, Aug 07 2013
Sum_{n>=0} 1/a(n) = (exp(1/4)*sqrt(Pi)*erf(1/2) + 2)/2 = 1 + A214869, where erf(x) is the error function. - Ilya Gutkovskiy, Nov 10 2016
Sum_{n>=0} (-1)^n/a(n) = 1 - sqrt(Pi)*erfi(1/2)/(2*exp(1/4)), where erfi(x) is the imaginary error function. - Amiram Eldar, Feb 20 2021
a(n) = 1/([x^n] hypergeom([1], [1/2], x/4)). - Peter Luschny, Sep 13 2024
a(n) = 2^n*n!*JacobiP(n, -1/2, -n, 3). - Peter Luschny, Jan 22 2025
G.f.: 2F0(1,1/2;;4x). - R. J. Mathar, Jun 07 2025

More terms from James Sellers, May 01 2000

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

Original entry on oeis.org

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)

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)

A052714 a(n) = 2^(n-1) * n! * Catalan(n-1) for n > 0 with a(0) = 0.

Original entry on oeis.org

0, 1, 4, 48, 960, 26880, 967680, 42577920, 2214051840, 132843110400, 9033331507200, 686533194547200, 57668788341964800, 5305528527460761600, 530552852746076160000, 57299708096576225280000, 6646766139202842132480000, 824199001261152424427520000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

a(n+1) is the number of square roots of any permutation in S_{8*n} whose disjoint cycle decomposition consists of 2*n cycles of length 4. - Luis Manuel Rivera Martínez, Feb 26 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 670.
Jesús Leaños, Rutilo Moreno and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, arXiv:1005.1531 [math.CO], 2010-2011.
Jesús Leaños, Rutilo Moreno and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, Australas. J. Combin., Vol. 52 (2012), pp. 41-54 (Theorem 1).

Sequences of the form m^(n-1)*n!*Catalan(n-1): A001813 (m=1), this sequence (or A144828) (m=2), A221954 (m=3), A052734 (m=4), A221953 (m=5), A221955 (m=6).

Cf. A000108, A221953, A221954.

[0] cat [Catalan(n-1)*2^(n-1)*Factorial(n): n in [1..20]]; // Vincenzo Librandi, Mar 11 2013

spec := [S,{B=Union(Z,C),S=Union(B,C),C=Prod(S,S)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Join[{0}, Table[CatalanNumber[n-1] 2^(n-1) n!, {n,  1, 20}]] (* Vincenzo Librandi, Mar 11 2013 *)

PARI
```
a(n)=if(n<1,0,2^(n-1)*(2*n-2)!/(n-1)!)
```

[0]+[2^(n-1)*factorial(n)*catalan_number(n-1) for n in (1..30)] # G. C. Greubel, Apr 02 2021

E.g.f.: (1- sqrt(1-8*x))/4.
Recurrence: a(1) = 1, 4*(1 - 2*n)*a(n) + a(n+1) = 0.
a(n) = A052701(n)*n!.
a(n) = 8^(n-1)*Gamma(n-1/2)/Pi^(1/2), n>0.
a(n+1) = A090802(2n, n). - Ross La Haye, Oct 18 2005
a(n) = 2^(n-1)*(2*n-2)!/(n-1)! for n>=1.
E.g.f. A(x) satisfies differential equation A'(x)=1/(1-4*A(x)). - Vladimir Kruchinin, May 04 2011
G.f.: x/(1-4x/(1-8x/(1-12x/(1-16x/(1-20x/(1-24x/(1-28x/(1-32x/(1-... (continued fraction). - Philippe Deléham, Jan 07 2012
G.f.: 2*x/G(0), where G(k)= 1 + 1/(1 - 2*x*(8*k+4)/(2*x*(8*k+4) - 1 + 16*x*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 30 2013
a(0) = 0, a(1) = 1; a(n) = 2 * Sum_{k=1..n-1} binomial(n,k) * a(k) * a(n-k). - Ilya Gutkovskiy, Jul 09 2020
From Amiram Eldar, Jan 08 2022: (Start)
Sum_{n>=1} 1/a(n) = 1 + e^(1/8)*sqrt(Pi)*erf(1/(2*sqrt(2)))/(2*sqrt(2)), where erf is the error function.
Sum_{n>=1} (-1)^(n+1)/a(n) = 1 - e^(-1/8)*sqrt(Pi)*erfi(1/(2*sqrt(2)))/(2*sqrt(2)), where erfi is the imaginary error function. (End)

Edited by N. J. A. Sloane, Feb 03 2013

A052734 a(n) = 4^(n-1) * n! * Catalan(n-1) for n > 0, with a(0) = 0.

Original entry on oeis.org

0, 1, 8, 192, 7680, 430080, 30965760, 2724986880, 283398635520, 34007836262400, 4625065731686400, 703009991216332800, 118105678524343910400, 21731444848479279513600, 4346288969695855902720000, 938798417454304874987520000, 217801232849398730997104640000, 54014705746650885287281950720000, 14259882317115833715842434990080000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

For n>0, the number of fully-parenthesized expressions that you can form with n operands and 4 types of binary operators. - Yogy Namara (yogy.namara(AT)gmail.com), Mar 07 2010

a(n+1) is the number of square roots of any permutation in S_{16*n} whose disjoint cycle decomposition consists of 2*n cycles of length 8. - Luis Manuel Rivera Martínez, Feb 26 2015

			Let's say the 4 types of binary operators are +, -, *, and /. Then, with 3 operands {a, b, c}, we can form expressions such as ((b+a)/c), (a-(c-b)), (c*(b+a)), etc. There are a(3)=192 such expressions. - Yogy Namara (yogy.namara(AT)gmail.com), Mar 07 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 690.
Jesús Leaños, Rutilo Moreno and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, arXiv:1005.1531 [math.CO], 2010-2011.
Jesús Leaños, Rutilo Moreno and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, Australas. J. Combin., Vol. 52 (2012), pp. 41-54 (Theorem 1).
W. van der Aalst, J. Buijs and B. van Dongen, Towards Improving the Representational Bias of Process Mining, 2012.

Sequences of the form m^(n-1)*n!*Catalan(n-1): A001813 (m=1), A052714 (or A144828) (m=2), A221954 (m=3), this sequence (m=4), A221953 (m=5), A221955 (m=6).

Equal to A000108 if all operands and all operators are indistinguishable.

[0] cat [Catalan(n-1)*4^(n-1)*Factorial(n): n in [1..20]]; // Vincenzo Librandi, Mar 11 2013

spec := [S,{B=Prod(C,C),S=Union(B,Z),C=Union(B,S,Z)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
seq((2*n)!/n! * 4^n, n = 0..10);

Join[{0}, Table[CatalanNumber[n-1] 4^(n-1) n!, {n, 1, 20}]] (* Vincenzo Librandi, Mar 11 2013 *)

[0]+[4^(n-1)*factorial(n)*catalan_number(n-1) for n in (1..30)] # G. C. Greubel, Apr 02 2021

E.g.f.: (1 - sqrt(1-16*x))/8.
Recurrence: a(1)=1, 8*(1 - 2*n)*a(n) + a(n+1) = 0.
a(n) = 16^n*Gamma(n+1/2)/sqrt(Pi).
a(0) = 0, a(1) = 1; a(n) = 4 * Sum_{k=1..n-1} binomial(n,k) * a(k) * a(n-k). - Ilya Gutkovskiy, Jul 09 2020
From Amiram Eldar, Jan 08 2022: (Start)
Sum_{n>=1} 1/a(n) = 1 + e^(1/16)*sqrt(Pi)*erf(1/4)/4, where erf is the error function.
Sum_{n>=1} (-1)^(n+1)/a(n) = 1 - e^(-1/16)*sqrt(Pi)*erfi(1/4)/4, where erfi is the imaginary error function. (End)

Entry revised by N. J. A. Sloane, Feb 04 2013 and Feb 06 2013

A221954 a(n) = 3^(n-1) * n! * Catalan(n-1).

Original entry on oeis.org

1, 6, 108, 3240, 136080, 7348320, 484989120, 37829151360, 3404623622400, 347271609484800, 39588963481267200, 4988209398639667200, 688372897012274073600, 103255934551841111040000, 16727461397398259988480000, 2910578283147297237995520000, 541367560665397286267166720000, 107190777011748662680899010560000, 22510063172467219162988792217600000

Offset: 1

N. J. A. Sloane, Feb 03 2013

nonn

a(n+1) is the number of square roots of any permutation in S_{12*n} whose disjoint cycle decomposition consists of 2*n cycles of length 6. - Luis Manuel Rivera Martínez, Feb 26 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..200
W. van der Aalst, J. Buijs and B. van Dongen, Towards Improving the Representational Bias of Process Mining, 2012.
Jesús Leaños, Rutilo Moreno and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, arXiv:1005.1531 [math.CO], 2010-2011.
Jesús Leaños, Rutilo Moreno and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, Australas. J. Combin., Vol. 52 (2012), pp. 41-54 (Theorem 1).

Sequences of the form m^(n-1)*n!*Catalan(n-1): A001813 (m=1), A052714 (or A144828) (m=2), this sequence (m=3), A052734 (m=4), A221953 (m=5), A221955 (m=6).

Cf. A000108.

[Catalan(n-1)*3^(n-1)*Factorial(n): n in [1..20]]; // Vincenzo Librandi, Mar 11 2013

A221954:= n-> (3^(n-1)*n!/(2*(2*n-1))*binomial(2*n,n); seq(A221954(n), n=1..30); # G. C. Greubel, Apr 02 2021

Table[CatalanNumber[n-1] 3^(n-1) n!, {n, 20}] (* Vincenzo Librandi, Mar 11 2013 *)

my(x='x+O('x^22)); Vec(serlaplace((1-sqrt(1-12*x))/6)) \\ Michel Marcus, Mar 04 2015

[3^(n-1)*factorial(n)*catalan_number(n-1) for n in (1..30)] # G. C. Greubel, Apr 02 2021

a(n) = 6*(2*n-3)*a(n-1) with a(1)=1. - Bruno Berselli, Mar 11 2013
E.g.f.: (1 - sqrt(1-12*x))/6. - Luis Manuel Rivera Martínez, Mar 04 2015
a(n) = 12^(n-1) * Gamma(n - 1/2) / sqrt(Pi). - Daniel Suteu, Jan 06 2017
a(1) = 1; a(n) = 3 * Sum_{k=1..n-1} binomial(n,k) * a(k) * a(n-k). - Ilya Gutkovskiy, Jul 09 2020
From Amiram Eldar, Jan 08 2022: (Start)
Sum_{n>=1} 1/a(n) = 1 + e^(1/12)*sqrt(Pi)*erf(1/(2*sqrt(3)))/(2*sqrt(3)), where erf is the error function.
Sum_{n>=1} (-1)^(n+1)/a(n) = 1 - e^(-1/12)*sqrt(Pi)*erfi(1/(2*sqrt(3)))/(2*sqrt(3)), where erfi is the imaginary error function. (End)

A257618 Triangle read by rows: 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) = 8*x + 2.

Original entry on oeis.org

1, 2, 2, 4, 40, 4, 8, 472, 472, 8, 16, 4928, 16992, 4928, 16, 32, 49824, 433984, 433984, 49824, 32, 64, 499584, 9505728, 22567168, 9505728, 499584, 64, 128, 4999040, 192085632, 909941120, 909941120, 192085632, 4999040, 128

Offset: 0

Dale Gerdemann, May 09 2015

			Triangle begins as:
    1;
    2,       2;
    4,      40,         4;
    8,     472,       472,         8;
   16,    4928,     16992,      4928,        16;
   32,   49824,    433984,    433984,     49824,        32;
   64,  499584,   9505728,  22567168,   9505728,    499584,      64;
  128, 4999040, 192085632, 909941120, 909941120, 192085632, 4999040, 128;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000079, A144828 (row sums), A167884.
Cf. A038208, A256890, A257609, A257610, A257612, A257614, A257616, A257617, A257619.
Similar sequences listed in A256890.

T[n_, k_, a_, b_]:= T[n, k, a, b]= If[k<0 || k>n, 0, If[n==0, 1, (a*(n-k)+b)*T[n-1, k-1, a, b] + (a*k+b)*T[n-1, k, a, b]]];
Table[T[n,k,8,2], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 24 2022 *)

def T(n,k,a,b): # A257618
    if (k<0 or k>n): return 0
    elif (n==0): return 1
    else: return  (a*k+b)*T(n-1,k,a,b) + (a*(n-k)+b)*T(n-1,k-1,a,b)
flatten([[T(n,k,8,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 24 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) = 8*x + 2.
Sum_{k=0..n} T(n, k) = A144828(n).
From G. C. Greubel, Mar 24 2022: (Start)
T(n, k) = (a*k + b)*T(n-1, k) + (a*(n-k) + b)*T(n-1, k-1), with T(n, 0) = 1, a = 8, and b = 2.
T(n, n-k) = T(n, k).
T(n, 0) = A000079(n).
T(n, 1) = 2^(n-1)*(5^n - 2*n - 1).
T(n, 2) = 2^(n-3)*(3^(2*n+1) -2*(2*n+1)*5^n -1 +4*n^2). (End)

A221953 a(n) = 5^(n-1) * n! * Catalan(n-1).

Original entry on oeis.org

1, 10, 300, 15000, 1050000, 94500000, 10395000000, 1351350000000, 202702500000000, 34459425000000000, 6547290750000000000, 1374931057500000000000, 316234143225000000000000, 79058535806250000000000000, 21345804667687500000000000000, 6190283353629375000000000000000, 1918987839625106250000000000000000

Offset: 1

N. J. A. Sloane, Feb 03 2013

nonn

a(n+1) is the number of square roots of any permutation in S_{20*n} whose disjoint cycle decomposition consists of 2*n cycles of length 10. - Luis Manuel Rivera Martínez, Feb 26 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..200
W. van der Aalst, J. Buijs and B. van Dongen, Towards Improving the Representational Bias of Process Mining, 2012.
Jesús Leaños, Rutilo Moreno and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, arXiv:1005.1531 [math.CO], 2010-2011.
Jesús Leaños, Rutilo Moreno and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, Australas. J. Combin., Vol. 52 (2012), pp. 41-54 (Theorem 1).

Sequences of the form m^(n-1)*n!*Catalan(n-1): A001813 (m=1), A052714 (or A144828) (m=2), A221954 (m=3), A052734 (m=4), this sequence (m=5), A221955 (m=6).

Cf. A000108.

[Catalan(n-1)*5^(n-1)*Factorial(n): n in [1..20]]; // Vincenzo Librandi, Mar 11 2013

A221953:= n-> (5^(n-1)*n!/(2*(2*n-1))*binomial(2*n,n); seq(A221953(n), n=1..30); # G. C. Greubel, Apr 02 2021

Table[CatalanNumber[n - 1]  5^(n-1)  n!, {n, 20}] (* Vincenzo Librandi, Mar 11 2013 *)

my(x='x+O('x^22)); Vec(serlaplace((1-sqrt(1-20*x))/10)) \\ Michel Marcus, Mar 04 2015

[5^(n-1)*factorial(n)*catalan_number(n-1) for n in (1..30)] # G. C. Greubel, Apr 02 2021

a(n) = 10*(2*n-3)*a(n-1) with a(1)=1. - Bruno Berselli, Mar 11 2013
E.g.f.: (1 - sqrt(1-20*x))/10. - Luis Manuel Rivera Martínez, Mar 04 2015
a(1) = 1; a(n) = 5 * Sum_{k=1..n-1} binomial(n,k) * a(k) * a(n-k). - Ilya Gutkovskiy, Jul 10 2020
From Amiram Eldar, Jan 08 2022: (Start)
Sum_{n>=1} 1/a(n) = 1 + e^(1/20)*sqrt(Pi)*erf(1/(2*sqrt(5)))/(2*sqrt(5)), where erf is the error function.
Sum_{n>=1} (-1)^(n+1)/a(n) = 1 - e^(-1/20)*sqrt(Pi)*erfi(1/(2*sqrt(5)))/(2*sqrt(5)), where erfi is the imaginary error function. (End)

A221955 a(n) = 6^(n-1) * n! * Catalan(n-1).

Original entry on oeis.org

1, 12, 432, 25920, 2177280, 235146240, 31039303680, 4842131374080, 871583647334400, 177803064056217600, 40539098604817612800, 10215852848414038425600, 2819575386162274605465600, 845872615848682381639680000, 274062727534973091651256320000, 95373829182170635894637199360000, 35479064455767476552805038161920000

Offset: 1

N. J. A. Sloane, Feb 03 2013

nonn

a(n+1) is the number of square roots of any permutation in S_{24*n} whose disjoint cycle decomposition consists of 2*n cycles of length 12. - Luis Manuel Rivera Martínez, Feb 28 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..200
W. van der Aalst, J. Buijs and B. van Dongen, Towards Improving the Representational Bias of Process Mining, 2012.
Jesús Leaños, Rutilo Moreno and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, Australas. J. Combin., Vol. 52 (2012), pp. 41-54 (Theorem 1).

Sequences of the form m^(n-1)*n!*Catalan(n-1): A001813 (m=1), A052714 (or A144828) (m=2), A221954 (m=3), A052734 (m=4), A221953 (m=5), this sequence (m=6).

Cf. A000108.

[Catalan(n-1)*6^(n-1)*Factorial(n): n in [1..20]]; // Vincenzo Librandi, Mar 11 2013

A221955:= n-> 3*6^(n-2)*n!*binomial(2*n,n)/(2*n-1); seq(A221955(n), n=1..30); # G. C. Greubel, Apr 02 2021

Table[CatalanNumber[n-1] 6^(n-1) n!, {n, 20}] (* Vincenzo Librandi, Mar 11 2013 *)
nxt[{n_,a_}]:={n+1,12a(2n-1)}; NestList[nxt,{1,1},20][[;;,2]] (* Harvey P. Dale, Sep 21 2024 *)

my(x='x+O('x^22)); Vec(serlaplace((1-sqrt(1-24*x))/12)) \\ Michel Marcus, Mar 04 2015

[6^(n-1)*factorial(n)*catalan_number(n-1) for n in (1..30)] # G. C. Greubel, Apr 02 2021

a(n) = 12*(2*n-3)*a(n-1) with a(1)=1. - Bruno Berselli, Mar 11 2013
E.g.f.: (1-sqrt(1-24*x))/12. - Luis Manuel Rivera Martínez, Mar 04 2015
a(1) = 1; a(n) = 6 * Sum_{k=1..n-1} binomial(n,k) * a(k) * a(n-k). - Ilya Gutkovskiy, Jul 10 2020
From Amiram Eldar, Jan 08 2022: (Start)
Sum_{n>=1} 1/a(n) = 1 + e^(1/24)*sqrt(Pi)*erf(1/(2*sqrt(6)))/(2*sqrt(6)), where erf is the error function.
Sum_{n>=1} (-1)^(n+1)/a(n) = 1 - e^(-1/24)*sqrt(Pi)*erfi(1/(2*sqrt(6)))/(2*sqrt(6)), where erfi is the imaginary error function. (End)

cp's OEIS Frontend

A001813 Quadruple factorial numbers: a(n) = (2n)!/n!.

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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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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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A052714 a(n) = 2^(n-1) * n! * Catalan(n-1) for n > 0 with a(0) = 0.

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A052734 a(n) = 4^(n-1) * n! * Catalan(n-1) for n > 0, with a(0) = 0.

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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.

A257618 Triangle read by rows: 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) = 8*x + 2.