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%I A000142 M1675 N0659 #811 May 03 2024 15:17:05
%S A000142 1,1,2,6,24,120,720,5040,40320,362880,3628800,39916800,479001600,
%T A000142 6227020800,87178291200,1307674368000,20922789888000,355687428096000,
%U A000142 6402373705728000,121645100408832000,2432902008176640000,51090942171709440000,1124000727777607680000
%N A000142 Factorial numbers: n! = 1*2*3*4*...*n (order of symmetric group S_n, number of permutations of n letters).
%C A000142 The earliest publication that discusses this sequence appears to be the Sepher Yezirah [Book of Creation], circa AD 300. (See Knuth, also the Zeilberger link.) - _N. J. A. Sloane_, Apr 07 2014
%C A000142 For n >= 1, a(n) is the number of n X n (0,1) matrices with each row and column containing exactly one entry equal to 1.
%C A000142 This sequence is the BinomialMean transform of A000354. (See A075271 for definition.) - _John W. Layman_, Sep 12 2002 [This is easily verified from the Paul Barry formula for A000354, by interchanging summations and using the formula: Sum_k (-1)^k C(n-i, k) = KroneckerDelta(i,n). - _David Callan_, Aug 31 2003]
%C A000142 Number of distinct subsets of T(n-1) elements with 1 element A, 2 elements B, ..., n - 1 elements X (e.g., at n = 5, we consider the distinct subsets of ABBCCCDDDD and there are 5! = 120). - _Jon Perry_, Jun 12 2003
%C A000142 n! is the smallest number with that prime signature. E.g., 720 = 2^4 * 3^2 * 5. - _Amarnath Murthy_, Jul 01 2003
%C A000142 a(n) is the permanent of the n X n matrix M with M(i, j) = 1. - _Philippe Deléham_, Dec 15 2003
%C A000142 Given n objects of distinct sizes (e.g., areas, volumes) such that each object is sufficiently large to simultaneously contain all previous objects, then n! is the total number of essentially different arrangements using all n objects. Arbitrary levels of nesting of objects are permitted within arrangements. (This application of the sequence was inspired by considering leftover moving boxes.) If the restriction exists that each object is able or permitted to contain at most one smaller (but possibly nested) object at a time, the resulting sequence begins 1,2,5,15,52 (Bell Numbers?). Sets of nested wooden boxes or traditional nested Russian dolls come to mind here. - _Rick L. Shepherd_, Jan 14 2004
%C A000142 From _Michael Somos_, Mar 04 2004; edited by _M. F. Hasler_, Jan 02 2015: (Start)
%C A000142 Stirling transform of [2, 2, 6, 24, 120, ...] is A052856 = [2, 2, 4, 14, 76, ...].
%C A000142 Stirling transform of [1, 2, 6, 24, 120, ...] is A000670 = [1, 3, 13, 75, ...].
%C A000142 Stirling transform of [0, 2, 6, 24, 120, ...] is A052875 = [0, 2, 12, 74, ...].
%C A000142 Stirling transform of [1, 1, 2, 6, 24, 120, ...] is A000629 = [1, 2, 6, 26, ...].
%C A000142 Stirling transform of [0, 1, 2, 6, 24, 120, ...] is A002050 = [0, 1, 5, 25, 140, ...].
%C A000142 Stirling transform of (A165326*A089064)(1...) = [1, 0, 1, -1, 8, -26, 194, ...] is [1, 1, 2, 6, 24, 120, ...] (this sequence). (End)
%C A000142 First Eulerian transform of 1, 1, 1, 1, 1, 1... The first Eulerian transform transforms a sequence s to a sequence t by the formula t(n) = Sum_{k=0..n} e(n, k)s(k), where e(n, k) is a first-order Eulerian number [A008292]. - _Ross La Haye_, Feb 13 2005
%C A000142 Conjecturally, 1, 6, and 120 are the only numbers which are both triangular and factorial. - Christopher M. Tomaszewski (cmt1288(AT)comcast.net), Mar 30 2005
%C A000142 n! is the n-th finite difference of consecutive n-th powers. E.g., for n = 3, [0, 1, 8, 27, 64, ...] -> [1, 7, 19, 37, ...] -> [6, 12, 18, ...] -> [6, 6, ...]. - Bryan Jacobs (bryanjj(AT)gmail.com), Mar 31 2005
%C A000142 a(n+1) = (n+1)! = 1, 2, 6, ... has e.g.f. 1/(1-x)^2. - _Paul Barry_, Apr 22 2005
%C A000142 Write numbers 1 to n on a circle. Then a(n) = sum of the products of all n - 2 adjacent numbers. E.g., a(5) = 1*2*3 + 2*3*4 + 3*4*5 + 4*5*1 +5*1*2 = 120. - _Amarnath Murthy_, Jul 10 2005
%C A000142 The number of chains of maximal length in the power set of {1, 2, ..., n} ordered by the subset relation. - _Rick L. Shepherd_, Feb 05 2006
%C A000142 The number of circular permutations of n letters for n >= 0 is 1, 1, 1, 2, 6, 24, 120, 720, 5040, 40320, ... - Xavier Noria (fxn(AT)hashref.com), Jun 04 2006
%C A000142 a(n) is the number of deco polyominoes of height n (n >= 1; see definitions in the Barcucci et al. references). - _Emeric Deutsch_, Aug 07 2006
%C A000142 a(n) is the number of partition tableaux of size n. See Steingrimsson/Williams link for the definition. - _David Callan_, Oct 06 2006
%C A000142 Consider the n! permutations of the integer sequence [n] = 1, 2, ..., n. The i-th permutation consists of ncycle(i) permutation cycles. Then, if the Sum_{i=1..n!} 2^ncycle(i) runs from 1 to n!, we have Sum_{i=1..n!} 2^ncycle(i) = (n+1)!. E.g., for n = 3 we have ncycle(1) = 3, ncycle(2) = 2, ncycle(3) = 1, ncycle(4) = 2, ncycle(5) = 1, ncycle(6) = 2 and 2^3 + 2^2 + 2^1 + 2^2 + 2^1 + 2^2 = 8 + 4 + 2 + 4 + 2 + 4 = 24 = (n+1)!. - _Thomas Wieder_, Oct 11 2006
%C A000142 a(n) is the number of set partitions of {1, 2, ..., 2n - 1, 2n} into blocks of size 2 (perfect matchings) in which each block consists of one even and one odd integer. For example, a(3) = 6 counts 12-34-56, 12-36-45, 14-23-56, 14-25-36, 16-23-45, 16-25-34. - _David Callan_, Mar 30 2007
%C A000142 Consider the multiset M = [1, 2, 2, 3, 3, 3, 4, 4, 4, 4, ...] = [1, 2, 2, ..., n x 'n'] and form the set U (where U is a set in the strict sense) of all subsets N (where N may be a multiset again) of M. Then the number of elements |U| of U is equal to (n+1)!. E.g. for M = [1, 2, 2] we get U = [[], [2], [2, 2], [1], [1, 2], [1, 2, 2]] and |U| = 3! = 6. This observation is a more formal version of the comment given already by _Rick L. Shepherd_, Jan 14 2004. - _Thomas Wieder_, Nov 27 2007
%C A000142 For n >= 1, a(n) = 1, 2, 6, 24, ... are the positions corresponding to the 1's in decimal expansion of Liouville's constant (A012245). - _Paul Muljadi_, Apr 15 2008
%C A000142 Triangle A144107 has n! for row sums (given n > 0) with right border n! and left border A003319, the INVERTi transform of (1, 2, 6, 24, ...). - _Gary W. Adamson_, Sep 11 2008
%C A000142 Equals INVERT transform of A052186 and row sums of triangle A144108. - _Gary W. Adamson_, Sep 11 2008
%C A000142 From _Abdullahi Umar_, Oct 12 2008: (Start)
%C A000142 a(n) is also the number of order-decreasing full transformations (of an n-chain).
%C A000142 a(n-1) is also the number of nilpotent order-decreasing full transformations (of an n-chain). (End)
%C A000142 n! is also the number of optimal broadcast schemes in the complete graph K_{n}, equivalent to the number of binomial trees embedded in K_{n} (see Calin D. Morosan, Information Processing Letters, 100 (2006), 188-193). - Calin D. Morosan (cd_moros(AT)alumni.concordia.ca), Nov 28 2008
%C A000142 Let S_{n} denote the n-star graph. The S_{n} structure consists of n S_{n-1} structures. This sequence gives the number of edges between the vertices of any two specified S_{n+1} structures in S_{n+2} (n >= 1). - _K.V.Iyer_, Mar 18 2009
%C A000142 Chromatic invariant of the sun graph S_{n-2}.
%C A000142 It appears that a(n+1) is the inverse binomial transform of A000255. - Timothy Hopper (timothyhopper(AT)hotmail.co.uk), Aug 20 2009
%C A000142 a(n) is also the determinant of a square matrix, An, whose coefficients are the reciprocals of beta function: a{i, j} = 1/beta(i, j), det(An) = n!. - _Enrique Pérez Herrero_, Sep 21 2009
%C A000142 The asymptotic expansions of the exponential integrals E(x, m = 1, n = 1) ~ exp(-x)/x*(1 - 1/x + 2/x^2 - 6/x^3 + 24/x^4 + ...) and E(x, m = 1, n = 2) ~ exp(-x)/x*(1 - 2/x + 6/x^2 - 24/x^3 + ...) lead to the factorial numbers. See A163931 and A130534 for more information. - _Johannes W. Meijer_, Oct 20 2009
%C A000142 Satisfies A(x)/A(x^2), A(x) = A173280. - _Gary W. Adamson_, Feb 14 2010
%C A000142 a(n) = G^n where G is the geometric mean of the first n positive integers. - _Jaroslav Krizek_, May 28 2010
%C A000142 Increasing colored 1-2 trees with choice of two colors for the rightmost branch of nonleaves. - _Wenjin Woan_, May 23 2011
%C A000142 Number of necklaces with n labeled beads of 1 color. - _Robert G. Wilson v_, Sep 22 2011
%C A000142 The sequence 1!, (2!)!, ((3!)!)!, (((4!)!)!)!, ..., ((...(n!)!)...)! (n times) grows too rapidly to have its own entry. See Hofstadter.
%C A000142 The e.g.f. of 1/a(n) = 1/n! is BesselI(0, 2*sqrt(x)). See Abramowitz-Stegun, p. 375, 9.3.10. - _Wolfdieter Lang_, Jan 09 2012
%C A000142 a(n) is the length of the n-th row which is the sum of n-th row in triangle A170942. - _Reinhard Zumkeller_, Mar 29 2012
%C A000142 Number of permutations of elements 1, 2, ..., n + 1 with a fixed element belonging to a cycle of length r does not depend on r and equals a(n). - _Vladimir Shevelev_, May 12 2012
%C A000142 a(n) is the number of fixed points in all permutations of 1, ..., n: in all n! permutations, 1 is first exactly (n-1)! times, 2 is second exactly (n-1)! times, etc., giving (n-1)!*n = n!. - _Jon Perry_, Dec 20 2012
%C A000142 For n >= 1, a(n-1) is the binomial transform of A000757. See Moreno-Rivera. - _Luis Manuel Rivera Martínez_, Dec 09 2013
%C A000142 Each term is divisible by its digital root (A010888). - _Ivan N. Ianakiev_, Apr 14 2014
%C A000142 For m >= 3, a(m-2) is the number hp(m) of acyclic Hamiltonian paths in a simple graph with m vertices, which is complete except for one missing edge. For m < 3, hp(m)=0. - _Stanislav Sykora_, Jun 17 2014
%C A000142 a(n) is the number of increasing forests with n nodes. - _Brad R. Jones_, Dec 01 2014
%C A000142 The factorial numbers can be calculated by means of the recurrence n! = (floor(n/2)!)^2 * sf(n) where sf(n) are the swinging factorials A056040. This leads to an efficient algorithm if sf(n) is computed via prime factorization. For an exposition of this algorithm see the link below. - _Peter Luschny_, Nov 05 2016
%C A000142 Treeshelves are ordered (plane) binary (0-1-2) increasing trees where the nodes of outdegree 1 come in 2 colors. There are n! treeshelves of size n, and classical Françon's bijection maps bijectively treeshelves into permutations. - _Sergey Kirgizov_, Dec 26 2016
%C A000142 Satisfies Benford's law [Diaconis, 1977; Berger-Hill, 2017] - _N. J. A. Sloane_, Feb 07 2017
%C A000142 a(n) = Sum((d_p)^2), where d_p is the number of standard tableaux in the Ferrers board of the integer partition p and summation is over all integer partitions p of n. Example: a(3) = 6. Indeed, the partitions of 3 are [3], [2,1], and [1,1,1], having 1, 2, and 1 standard tableaux, respectively; we have 1^2 + 2^2 + 1^2 = 6. - _Emeric Deutsch_, Aug 07 2017
%C A000142 a(n) is the n-th derivative of x^n. - _Iain Fox_, Nov 19 2017
%C A000142 a(n) is the number of maximum chains in the n-dimensional Boolean cube {0,1}^n in respect to the relation "precedes". It is defined as follows: for arbitrary vectors u, v of {0,1}^n, such that u = (u_1, u_2, ..., u_n) and v = (v_1, v_2, ..., v_n), "u precedes v" if u_i <= v_i, for i=1, 2, ..., n. - _Valentin Bakoev_, Nov 20 2017
%C A000142 a(n) is the number of shortest paths (for example, obtained by Breadth First Search) between the nodes (0,0,...,0) (i.e., the all-zeros vector) and (1,1,...,1) (i.e., the all-ones vector) in the graph H_n, corresponding to the n-dimensional Boolean cube {0,1}^n. The graph is defined as H_n = (V_n, E_n), where V_n is the set of all vectors of {0,1}^n, and E_n contains edges formed by each pair adjacent vectors. - _Valentin Bakoev_, Nov 20 2017
%C A000142 a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = sigma(gcd(i,j)) for 1 <= i,j <= n. - _Bernard Schott_, Dec 05 2018
%C A000142 a(n) is also the number of inversion sequences of length n. A length n inversion sequence e_1, e_2, ..., e_n is a sequence of n integers such that 0 <= e_i < i. - _Juan S. Auli_, Oct 14 2019
%C A000142 The term "factorial" ("factorielle" in French) was coined by the French mathematician Louis François Antoine Arbogast (1759-1803) in 1800. The notation "!" was first used by the French mathematician Christian Kramp (1760-1826) in 1808. - _Amiram Eldar_, Apr 16 2021
%C A000142 Also the number of signotopes of rank 2, i.e., mappings X:{{1..n} choose 2}->{+,-} such that for any three indices a < b < c, the sequence X(a,b), X(a,c), X(b,c) changes its sign at most once (see Felsner-Weil reference). - _Manfred Scheucher_, Feb 09 2022
%C A000142 a(n) is also the number of labeled commutative semisimple rings with n elements. As an example the only commutative semisimple rings with 4 elements are F_4 and F_2 X F_2. They both have exactly 2 automorphisms, hence a(4)=24/2+24/2=24. - _Paul Laubie_, Mar 05 2024
%C A000142 a(n) is the number of extremely unlucky Stirling permutations of order n+1; i.e., the number of Stirling permutations of order n+1 that have exactly one lucky car. - _Bridget Tenner_, Apr 09 2024
%D A000142 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 833.
%D A000142 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 125; also p. 90, ex. 3.
%D A000142 Diaconis, Persi, The distribution of leading digits and uniform distribution mod 1, Ann. Probability, 5, 1977, 72--81,
%D A000142 Douglas R. Hofstadter, Fluid concepts & creative analogies: computer models of the fundamental mechanisms of thought, Basic Books, 1995, pages 44-46.
%D A000142 A. N. Khovanskii. The Application of Continued Fractions and Their Generalizations to Problem in Approximation Theory. Groningen: Noordhoff, Netherlands, 1963. See p. 141 (10.19).
%D A000142 D. E. Knuth, The Art of Computer Programming, Vol. 3, Section 5.1.2, p. 23. [From _N. J. A. Sloane_, Apr 07 2014]
%D A000142 J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 693 pp. 90, 297, Ellipses Paris 2004.
%D A000142 A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992.
%D A000142 R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
%D A000142 Sepher Yezirah [Book of Creation], circa AD 300. See verse 52.
%D A000142 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000142 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A000142 D. Stanton and D. White, Constructive Combinatorics, Springer, 1986; see p. 91.
%D A000142 Carlo Suares, Sepher Yetsira, Shambhala Publications, 1976. See verse 52.
%D A000142 David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, pp. 102.
%H A000142 N. J. A. Sloane, The first 100 factorials: Table of n, n! for n = 0..100
%H A000142 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A000142 L. F. A. Arbogast, Du calcul des dérivations, Strasbourg: Levrault, 1800.
%H A000142 S. B. Akers and B. Krishnamurthy, A group-theoretic model for symmetric interconnection networks, IEEE Trans. Comput., 38(4), April 1989, 555-566.
%H A000142 Masanori Ando, Odd number and Trapezoidal number, arXiv:1504.04121 [math.CO], 2015.
%H A000142 David Applegate and N. J. A. Sloane, Table giving cycle index of S_0 through S_40 in Maple format [gzipped]
%H A000142 C. Banderier, M. Bousquet-Mélou, A. Denise, P. Flajolet, D. Gardy and D. Gouyou-Beauchamps, Generating Functions for Generating Trees, Discrete Mathematics 246(1-3), March 2002, pp. 29-55.
%H A000142 Stefano Barbero, Umberto Cerruti and Nadir Murru, On the operations of sequences in rings and binomial type sequences, Ricerche di Matematica (2018), pp 1-17., also arXiv:1805.11922 [math.NT], 2018.
%H A000142 E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
%H A000142 E. Barcucci, A. Del Lungo, R. Pinzani and R. Sprugnoli, La hauteur des polyominos dirigés verticalement convexes, Actes du 31e Séminaire Lotharingien de Combinatoire, Publ. IRMA, Université Strasbourg I (1993).
%H A000142 Jean-Luc Baril, Sergey Kirgizov and Vincent Vajnovszki, Patterns in treeshelves, Discrete Mathematics, Vol. 340, No. 12 (2017), 2946-2954, arXiv:1611.07793 [cs.DM], 2016.
%H A000142 A. Berger and T. P. Hill, What is Benford's Law?, Notices, Amer. Math. Soc., 64:2 (2017), 132-134.
%H A000142 M. Bhargava, The factorial function and generalizations, Amer. Math. Monthly, 107 (Nov. 2000), 783-799.
%H A000142 Natasha Blitvić and Einar Steingrímsson, Permutations, moments, measures, arXiv:2001.00280 [math.CO], 2020.
%H A000142 Henry Bottomley, Illustration of initial terms
%H A000142 Douglas Butler, Factorials!
%H A000142 David Callan, Counting Stabilized-Interval-Free Permutations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.1.8.
%H A000142 Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000142 Laura Colmenarejo, Aleyah Dawkins, Jennifer Elder, Pamela E. Harris, Kimberly J. Harry, Selvi Kara, Dorian Smith, and Bridget Eileen Tenner, On the lucky and displacement statistics of Stirling permutations, arXiv:2403.03280 [math.CO], 2024.
%H A000142 CombOS - Combinatorial Object Server, Generate permutations
%H A000142 Robert M. Dickau, Permutation diagrams
%H A000142 S. Felsner and H. Weil, Sweeps, arrangements and signotopes, Discrete Applied Mathematics Volume 109, Issues 1-2, 2001, Pages 67-94.
%H A000142 Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, 2009; see page 18
%H A000142 J. Françon, Arbres binaires de recherche : propriétés combinatoires et applications, Revue française d'automatique informatique recherche opérationnelle, Informatique théorique, 10 no. 3 (1976), pp. 35-50.
%H A000142 H. Fripertinger, The elements of the symmetric group
%H A000142 H. Fripertinger, The elements of the symmetric group in cycle notation
%H A000142 Joël Gay and Vincent Pilaud, The weak order on Weyl posets, arXiv:1804.06572 [math.CO], 2018.
%H A000142 Elizabeth Hartung, Hung Phuc Hoang, Torsten Mütze, and Aaron Williams, Combinatorial generation via permutation languages. I. Fundamentals, arXiv:1906.06069 [cs.DM], 2019.
%H A000142 A. M. Ibrahim, Extension of factorial concept to negative numbers, Notes on Number Theory and Discrete Mathematics, Vol. 19, 2013, 2, 30-42.
%H A000142 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 20
%H A000142 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 297
%H A000142 Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
%H A000142 Milan Janjic, Determinants and Recurrence Sequences, Journal of Integer Sequences, 2012, Article 12.3.5. - _N. J. A. Sloane_, Sep 16 2012
%H A000142 B. R. Jones, On tree hook length formulas, Feynman rules and B-series, p. 22, Master's thesis, Simon Fraser University, 2014.
%H A000142 Clark Kimberling, Matrix Transformations of Integer Sequences, J. Integer Seqs., Vol. 6, 2003.
%H A000142 Christian Kramp, Élémens d'arithmétique universelle, Cologne: De l'imprimerie de Th. F. Thiriart, 1808.
%H A000142 G. Labelle et al., Stirling numbers interpolation using permutations with forbidden subsequences, Discrete Math. 246 (2002), 177-195.
%H A000142 Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%H A000142 John W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
%H A000142 Paul Leyland, Generalized Cullen and Woodall numbers [Cached copy at the Wayback Machine]
%H A000142 Peter Luschny, Swing, divide and conquer the factorial, (excerpt).
%H A000142 Rutilo Moreno and Luis Manuel Rivera, Blocks in cycles and k-commuting permutations, arXiv:1306:5708 [math.CO], 2013-2014.
%H A000142 Thomas Morrill, Further Development of "Non-Pythagorean" Musical Scales Based on Logarithms, arXiv:1804.08067 [math.HO], 2018.
%H A000142 T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
%H A000142 Norihiro Nakashima and Shuhei Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019.
%H A000142 N. E. Nørlund, Vorlesungen über Differenzenrechnung Springer 1924, p. 98.
%H A000142 R. Ondrejka, 1273 exact factorials, Math. Comp., 24 (1970), 231.
%H A000142 Enrique Pérez Herrero, Beta function matrix determinant Psychedelic Geometry blogspot-09/21/09
%H A000142 Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7
%H A000142 M. Prunescu and L. Sauras-Altuzarra, An arithmetic term for the factorial function, Examples and Counterexamples, Vol. 5 (2024).
%H A000142 Fred Richman, Multiple precision arithmetic(Computing factorials up to 765!)
%H A000142 Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
%H A000142 David A. Sheppard, The factorial representation of major balanced labelled graphs, Discrete Math., 15(1976), no. 4, 379-388.
%H A000142 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.
%H A000142 R. P. Stanley, A combinatorial miscellany
%H A000142 R. P. Stanley, Recent Progress in Algebraic Combinatorics, Bull. Amer. Math. Soc., 40 (2003), 55-68.
%H A000142 Einar Steingrimsson and Lauren K. Williams, Permutation tableaux and permutation patterns, arXiv:math/0507149 [math.CO], 2005-2006.
%H A000142 A. Umar, On the semigroups of order-decreasing finite full transformations, Proc. Roy. Soc. Edinburgh 120A (1992), 129-142.
%H A000142 G. Villemin's Almanach of Numbers, Factorielles
%H A000142 Sage Weil, The First 999 Factorials [Cached copy at the Wayback Machine]
%H A000142 Eric Weisstein's World of Mathematics, Factorial, Gamma Function, Multifactorial, Permutation, Permutation Pattern, Laguerre Polynomial, Diagonal Matrix, Chromatic Invariant.
%H A000142 R. W. Whitty, Rook polynomials on two-dimensional surfaces and graceful labellings of graphs, Discrete Math., 308 (2008), 674-683.
%H A000142 Wikipedia, Factorial
%H A000142 Doron Zeilberger, King Solomon and Rabbi Ben Ezra's Evaluations of Pi and Patriarch Abraham's Analysis of an Algorithm.
%H A000142 Doron Zeilberger, King Solomon and Rabbi Ben Ezra's Evaluations of Pi and Patriarch Abraham's Analysis of an Algorithm [Local copy]
%H A000142 Doron Zeilberger and Noam Zeilberger, Two Questions about the Fractional Counting of Partitions, arXiv:1810.12701 [math.CO], 2018.
%H A000142 Index entries for "core" sequences
%H A000142 Index to divisibility sequences
%H A000142 Index entries for sequences related to factorial numbers
%H A000142 Index entries for sequences related to Benford's law
%F A000142 Exp(x) = Sum_{m >= 0} x^m/m!. - _Mohammad K. Azarian_, Dec 28 2010
%F A000142 Sum_{i=0..n} (-1)^i * i^n * binomial(n, i) = (-1)^n * n!. - Yong Kong (ykong(AT)curagen.com), Dec 26 2000
%F A000142 Sum_{i=0..n} (-1)^i * (n-i)^n * binomial(n, i) = n!. - Peter C. Heinig (algorithms(AT)gmx.de), Apr 10 2007
%F A000142 The sequence trivially satisfies the recurrence a(n+1) = Sum_{k=0..n} binomial(n,k) * a(k)*a(n-k). - _Robert FERREOL_, Dec 05 2009
%F A000142 D-finite with recurrence: a(n) = n*a(n-1), n >= 1. n! ~ sqrt(2*Pi) * n^(n+1/2) / e^n (Stirling's approximation).
%F A000142 a(0) = 1, a(n) = subs(x = 1, (d^n/dx^n)(1/(2-x))), n = 1, 2, ... - _Karol A. Penson_, Nov 12 2001
%F A000142 E.g.f.: 1/(1-x). - _Michael Somos_, Mar 04 2004
%F A000142 a(n) = Sum_{k=0..n} (-1)^(n-k)*A000522(k)*binomial(n, k) = Sum_{k=0..n} (-1)^(n-k)*(x+k)^n*binomial(n, k). - _Philippe Deléham_, Jul 08 2004
%F A000142 Binomial transform of A000166. - _Ross La Haye_, Sep 21 2004
%F A000142 a(n) = Sum_{i=1..n} ((-1)^(i-1) * sum of 1..n taken n - i at a time) - e.g., 4! = (1*2*3 + 1*2*4 + 1*3*4 + 2*3*4) - (1*2 + 1*3 + 1*4 + 2*3 + 2*4 + 3*4) + (1 + 2 + 3 + 4) - 1 = (6 + 8 + 12 + 24) - (2 + 3 + 4 + 6 + 8 + 12) + 10 - 1 = 50 - 35 + 10 - 1 = 24. - _Jon Perry_, Nov 14 2005
%F A000142 a(n) = (n-1)*(a(n-1) + a(n-2)), n >= 2. - Matthew J. White, Feb 21 2006
%F A000142 1 / a(n) = determinant of matrix whose (i,j) entry is (i+j)!/(i!(j+1)!) for n > 0. This is a matrix with Catalan numbers on the diagonal. - _Alexander Adamchuk_, Jul 04 2006
%F A000142 Hankel transform of A074664. - _Philippe Deléham_, Jun 21 2007
%F A000142 For n >= 2, a(n-2) = (-1)^n*Sum_{j=0..n-1} (j+1)*Stirling1(n,j+1). - _Milan Janjic_, Dec 14 2008
%F A000142 From _Paul Barry_, Jan 15 2009: (Start)
%F A000142 G.f.: 1/(1-x-x^2/(1-3x-4x^2/(1-5x-9x^2/(1-7x-16x^2/(1-9x-25x^2... (continued fraction), hence Hankel transform is A055209.
%F A000142 G.f. of (n+1)! is 1/(1-2x-2x^2/(1-4x-6x^2/(1-6x-12x^2/(1-8x-20x^2... (continued fraction), hence Hankel transform is A059332. (End)
%F A000142 a(n) = Product_{p prime} p^(Sum_{k > 0} floor(n/p^k)) by Legendre's formula for the highest power of a prime dividing n!. - _Jonathan Sondow_, Jul 24 2009
%F A000142 a(n) = A053657(n)/A163176(n) for n > 0. - _Jonathan Sondow_, Jul 26 2009
%F A000142 It appears that a(n) = (1/0!) + (1/1!)*n + (3/2!)*n*(n-1) + (11/3!)*n*(n-1)*(n-2) + ... + (b(n)/n!)*n*(n-1)*...*2*1, where a(n) = (n+1)! and b(n) = A000255. - _Timothy Hopper_, Aug 12 2009
%F A000142 Sum_{n >= 0} 1/a(n) = e. - _Jaume Oliver Lafont_, Mar 03 2009
%F A000142 a(n) = a(n-1)^2/a(n-2) + a(n-1), n >= 2. - _Jaume Oliver Lafont_, Sep 21 2009
%F A000142 a(n) = Gamma(n+1). - _Enrique Pérez Herrero_, Sep 21 2009
%F A000142 a(n) = A173333(n,1). - _Reinhard Zumkeller_, Feb 19 2010
%F A000142 a(n) = A_{n}(1) where A_{n}(x) are the Eulerian polynomials. - _Peter Luschny_, Aug 03 2010
%F A000142 a(n) = n*(2*a(n-1) - (n-1)*a(n-2)), n > 1. - _Gary Detlefs_, Sep 16 2010
%F A000142 1/a(n) = -Sum_{k=1..n+1} (-2)^k*(n+k+2)*a(k)/(a(2*k+1)*a(n+1-k)). - _Groux Roland_, Dec 08 2010
%F A000142 From _Vladimir Shevelev_, Feb 21 2011: (Start)
%F A000142 a(n) = Product_{p prime, p <= n} p^(Sum_{i >= 1} floor(n/p^i)).
%F A000142 The infinitary analog of this formula is: a(n) = Product_{q terms of A050376 <= n} q^((n)_q), where (n)_q denotes the number of those numbers <= n for which q is an infinitary divisor (for the definition see comment in A037445). (End)
%F A000142 The terms are the denominators of the expansion of sinh(x) + cosh(x). - _Arkadiusz Wesolowski_, Feb 03 2012
%F A000142 G.f.: 1 / (1 - x / (1 - x / (1 - 2*x / (1 - 2*x / (1 - 3*x / (1 - 3*x / ... )))))). - _Michael Somos_, May 12 2012
%F A000142 G.f. 1 + x/(G(0)-x) where G(k) = 1 - (k+1)*x/(1 - x*(k+2)/G(k+1)); (continued fraction, 2-step). - _Sergei N. Gladkovskii_, Aug 14 2012
%F A000142 G.f.: W(1,1;-x)/(W(1,1;-x) - x*W(1,2;-x)), where W(a,b,x) = 1 - a*b*x/1! + a*(a+1)*b*(b+1)*x^2/2! - ... + a*(a+1)*...*(a+n-1)*b*(b+1)*...*(b+n-1)*x^n/n! + ...; see [A. N. Khovanskii, p. 141 (10.19)]. - _Sergei N. Gladkovskii_, Aug 15 2012
%F A000142 From _Sergei N. Gladkovskii_, Dec 26 2012: (Start)
%F A000142 G.f.: A(x) = 1 + x/(G(0) - x) where G(k) = 1 + (k+1)*x - x*(k+2)/G(k+1); (continued fraction).
%F A000142 Let B(x) be the g.f. for A051296, then A(x) = 2 - 1/B(x). (End)
%F A000142 G.f.: 1 + x*(G(0) - 1)/(x-1) where G(k) = 1 - (2*k+1)/(1-x/(x - 1/(1 - (2*k+2)/(1-x/(x - 1/G(k+1) ))))); (continued fraction). - _Sergei N. Gladkovskii_, Jan 15 2013
%F A000142 G.f.: 1 + x*(1 - G(0))/(sqrt(x)-x) where G(k) = 1 - (k+1)*sqrt(x)/(1-sqrt(x)/(sqrt(x)-1/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jan 25 2013
%F A000142 G.f.: 1 + x/G(0) where G(k) = 1 - x*(k+2)/( 1 - x*(k+1)/G(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Mar 23 2013
%F A000142 a(n) = det(S(i+1, j), 1 <= i, j <=n ), where S(n,k) are Stirling numbers of the second kind. - _Mircea Merca_, Apr 04 2013
%F A000142 G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(k+1)/(x*(k+1) + 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 24 2013
%F A000142 G.f.: 2/G(0), where G(k) = 1 + 1/(1 - 1/(1 - 1/(2*x*(k+1)) + 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 29 2013
%F A000142 G.f.: G(0), where G(k) = 1 + x*(2*k+1)/(1 - x*(2*k+2)/(x*(2*k+2) + 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 07 2013
%F A000142 a(n) = P(n-1, floor(n/2)) * floor(n/2)! * (n - (n-2)*((n+1) mod 2)), where P(n, k) are the k-permutations of n objects, n > 0. - _Wesley Ivan Hurt_, Jun 07 2013
%F A000142 a(n) = a(n-2)*(n-1)^2 + a(n-1), n > 1. - _Ivan N. Ianakiev_, Jun 18 2013
%F A000142 a(n) = a(n-2)*(n^2-1) - a(n-1), n > 1. - _Ivan N. Ianakiev_, Jun 30 2013
%F A000142 G.f.: 1 + x/Q(0), m=+2, where Q(k) = 1 - 2*x*(2*k+1) - m*x^2*(k+1)*(2*k+1)/( 1 - 2*x*(2*k+2) - m*x^2*(k+1)*(2*k+3)/Q(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Sep 24 2013
%F A000142 a(n) = A245334(n,n). - _Reinhard Zumkeller_, Aug 31 2014
%F A000142 a(n) = Product_{i = 1..n} A014963^floor(n/i) = Product_{i = 1..n} A003418(floor(n/i)). - _Matthew Vandermast_, Dec 22 2014
%F A000142 a(n) = round(Sum_{k>=1} log(k)^n/k^2), for n>=1, which is related to the n-th derivative of the Riemann zeta function at x=2 as follows: round((-1)^n * zeta^(n)(2)). Also see A073002. - _Richard R. Forberg_, Dec 30 2014
%F A000142 a(n) ~ Sum_{j>=0} j^n/e^j, where e = A001113. When substituting a generic variable for "e" this infinite sum is related to Eulerian polynomials. See A008292. This approximation of n! is within 0.4% at n = 2. See A255169. Accuracy, as a percentage, improves rapidly for larger n. - _Richard R. Forberg_, Mar 07 2015
%F A000142 a(n) = Product_{k=1..n} (C(n+1, 2)-C(k, 2))/(2*k-1); see Masanori Ando link. - _Michel Marcus_, Apr 17 2015
%F A000142 Sum_{n>=0} a(n)/(a(n + 1)*a(n + 2)) = Sum_{n>=0} 1/((n + 2)*(n + 1)^2*a(n)) = 2 - exp(1) - gamma + Ei(1) = 0.5996203229953..., where gamma = A001620, Ei(1) = A091725. - _Ilya Gutkovskiy_, Nov 01 2016
%F A000142 a(2^n) = 2^(2^n - 1) * 1!! * 3!! * 7!! * ... * (2^n - 1)!!. For example, 16! = 2^15*(1*3)*(1*3*5*7)*(1*3*5*7*9*11*13*15) = 20922789888000. - _Peter Bala_, Nov 01 2016
%F A000142 a(n) = sum(prod(B)), where the sum is over all subsets B of {1,2,...,n-1} and where prod(B) denotes the product of all the elements of set B. If B is a singleton set with element b, then we define prod(B)=b, and, if B is the empty set, we define prod(B) to be 1. For example, a(4)=(1*2*3)+(1*2)+(1*3)+(2*3)+(1)+(2)+(3)+1=24. - _Dennis P. Walsh_, Oct 23 2017
%F A000142 Sum_{n >= 0} 1/(a(n)*(n+2)) = 1. - Multiplying the denominator by (n+2) in Jaume Oliver Lafont's entry above creates a telescoping sum. - _Fred Daniel Kline_, Nov 08 2020
%F A000142 O.g.f.: Sum_{k >= 0} k!*x^k = Sum_{k >= 0} (k+y)^k*x^k/(1 + (k+y)*x)^(k+1) for arbitrary y. - _Peter Bala_, Mar 21 2022
%F A000142 E.g.f.: 1/(1 + LambertW(-x*exp(-x))) = 1/(1-x), see A258773. -(1/x)*substitute(z = x*exp(-x), z*(d/dz)LambertW(-z)) = 1/(1 - x). See A075513. Proof: Use the compositional inverse (x*exp(-x))^[-1] = -LambertW(-z). See A000169 or A152917, and Richard P. Stanley: Enumerative Combinatorics, vol. 2, p. 37, eq. (5.52). - _Wolfdieter Lang_, Oct 17 2022
%F A000142 Sum_{k >= 1} 1/10^a(k) = A012245 (Liouville constant). - _Bernard Schott_, Dec 18 2022
%F A000142 From _David Ulgenes_, Sep 19 2023: (Start)
%F A000142 1/a(n) = (e/(2*Pi*n)*Integral_{x=-oo..oo} cos(x-n*arctan(x))/(1+x^2)^(n/2) dx). Proof: take the real component of Laplace's integral for 1/Gamma(x).
%F A000142 a(n) = Integral_{x=0..1} e^(-t)*LerchPhi(1/e, -n, t) dt. Proof: use the relationship Gamma(x+1) = Sum_{n >= 0} Integral_{t=n..n+1} e^(-t)t^x dt = Sum_{n >= 0} Integral_{t=0..1} e^(-(t+n))(t+n)^x dt and interchange the order of summation and integration.
%F A000142 Conjecture: a(n) = 1/(2*Pi)*Integral_{x=-oo..oo}(n+i*x+1)!/(i*x+1)-(n+i*x-1)!/(i*x-1)dx. (End)
%F A000142 a(n) = floor(b(n)^n / (floor(((2^b(n) + 1) / 2^n)^b(n)) mod 2^b(n))), where b(n) = (n + 1)^(n + 2) = A007778(n+1). Joint work with _Mihai Prunescu_. - _Lorenzo Sauras Altuzarra_, Oct 18 2023
%F A000142 a(n) = e^(Integral_{x=1..n+1} Psi(x) dx) where Psi(x) is the digamma function. - _Andrea Pinos_, Jan 10 2024
%F A000142 a(n) = Integral_{x=0..oo} e^(-x^(1/n)) dx, for n > 0. - _Ridouane Oudra_, Apr 20 2024
%e A000142 There are 3! = 1*2*3 = 6 ways to arrange 3 letters {a, b, c}, namely abc, acb, bac, bca, cab, cba.
%e A000142 Let n = 2. Consider permutations of {1, 2, 3}. Fix element 3. There are a(2) = 2 permutations in each of the following cases: (a) 3 belongs to a cycle of length 1 (permutations (1, 2, 3) and (2, 1, 3)); (b) 3 belongs to a cycle of length 2 (permutations (3, 2, 1) and (1, 3, 2)); (c) 3 belongs to a cycle of length 3 (permutations (2, 3, 1) and (3, 1, 2)). - _Vladimir Shevelev_, May 13 2012
%e A000142 G.f. = 1 + x + 2*x^2 + 6*x^3 + 24*x^4 + 120*x^5 + 720*x^6 + 5040*x^7 + ...
%p A000142 A000142 := n -> n!; seq(n!,n=0..20);
%p A000142 spec := [ S, {S=Sequence(Z) }, labeled ]; seq(combstruct[count](spec,size=n), n=0..20);
%p A000142 # Maple program for computing cycle indices of symmetric groups
%p A000142 M:=6: f:=array(0..M): f[0]:=1: print(`n= `,0); print(f[0]); f[1]:=x[1]: print(`n= `, 1); print(f[1]); for n from 2 to M do f[n]:=expand((1/n)*add( x[l]*f[n-l],l=1..n)); print(`n= `, n); print(f[n]); od:
%p A000142 with(combstruct):ZL0:=[S,{S=Set(Cycle(Z,card>0))},labeled]: seq(count(ZL0,size=n),n=0..20); # _Zerinvary Lajos_, Sep 26 2007
%t A000142 Table[Factorial[n], {n, 0, 20}] (* _Stefan Steinerberger_, Mar 30 2006 *)
%t A000142 FoldList[#1 #2 &, 1, Range@ 20] (* _Robert G. Wilson v_, May 07 2011 *)
%t A000142 Range[20]! (* _Harvey P. Dale_, Nov 19 2011 *)
%t A000142 RecurrenceTable[{a[n] == n*a[n - 1], a[0] == 1}, a, {n, 0, 22}] (* _Ray Chandler_, Jul 30 2015 *)
%o A000142 (Axiom) [factorial(n) for n in 0..10]
%o A000142 (Magma) a:= func< n | Factorial(n) >; [ a(n) : n in [0..10]];
%o A000142 (Haskell)
%o A000142 a000142 :: (Enum a, Num a, Integral t) => t -> a
%o A000142 a000142 n = product [1 .. fromIntegral n]
%o A000142 a000142_list = 1 : zipWith (*) [1..] a000142_list
%o A000142 -- _Reinhard Zumkeller_, Mar 02 2014, Nov 02 2011, Apr 21 2011
%o A000142 (Python)
%o A000142 for i in range(1, 1000):
%o A000142 y = i
%o A000142 for j in range(1, i):
%o A000142 y *= i - j
%o A000142 print(y, "\n")
%o A000142 (Python)
%o A000142 import math
%o A000142 for i in range(1, 1000):
%o A000142 math.factorial(i)
%o A000142 print("")
%o A000142 # _Ruskin Harding_, Feb 22 2013
%o A000142 (PARI) a(n)=prod(i=1, n, i) \\ _Felix Fröhlich_, Aug 17 2014
%o A000142 (PARI) {a(n) = if(n<0, 0, n!)}; /* _Michael Somos_, Mar 04 2004 */
%o A000142 (Sage) [factorial(n) for n in (1..22)] # _Giuseppe Coppoletta_, Dec 05 2014
%o A000142 (GAP) List([0..22],Factorial); # _Muniru A Asiru_, Dec 05 2018
%o A000142 (Scala) (1: BigInt).to(24: BigInt).scanLeft(1: BigInt)(_ * _) // _Alonso del Arte_, Mar 02 2019
%o A000142 (Julia) print([factorial(big(n)) for n in 0:28]) # _Paul Muljadi_, May 01 2024
%Y A000142 Cf. A000165, A001044, A001563, A003422, A009445, A010050, A012245, A033312, A034886, A038507, A047920, A048631.
%Y A000142 Factorial base representation: A007623.
%Y A000142 Cf. A003319, A052186, A144107, A144108. - _Gary W. Adamson_, Sep 11 2008
%Y A000142 Complement of A063992. - _Reinhard Zumkeller_, Oct 11 2008
%Y A000142 Cf. A053657, A163176. - _Jonathan Sondow_, Jul 26 2009
%Y A000142 Cf. A173280. - _Gary W. Adamson_, Feb 14 2010
%Y A000142 Boustrophedon transforms: A230960, A230961.
%Y A000142 Cf. A233589.
%Y A000142 Cf. A245334.
%Y A000142 A row of the array in A249026.
%Y A000142 Cf. A001013 (multiplicative closure).
%Y A000142 For factorials with initial digit d (1 <= d <= 9) see A045509, A045510, A045511, A045516, A045517, A045518, A282021, A045519; A045520, A045521, A045522, A045523, A045524, A045525, A045526, A045527, A045528, A045529.
%Y A000142 Cf. A000169, A075513, A152917, A258773.
%K A000142 core,easy,nonn,nice
%O A000142 0,3
%A A000142 _N. J. A. Sloane_
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