0,3

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.

Sum((-1)^i * i^n * binomial(n, i), i=0..n) = (-1)^n * n! - Yong Kong (ykong(AT)curagen.com), Dec 26 2000

Sum((-1)^i * (n-i)^n * binomial(n, i), i=0..n) = n! - Peter C. Heinig (algorithms(AT)gmx.de), Apr 10 2007

This sequence is the BinomialMean transform of A000354. (See A075271 for definition.) - John W. Layman (layman(AT)math.vt.edu), 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 (callan(AT)stat.wisc.edu), Aug 31 2003

Number of distinct subsets of T(n-1) elements with 1 element A, 2 elements B,..., n-1 elements X (e.g. n=5, we consider the distinct subsets of ABBCCCDDDD, and there are 5!=120.) - Jon Perry (perry(AT)globalnet.co.uk), Jun 12 2003

n! is the smallest number with that prime signature. E.g. 720 = 2^4*3^2*5. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jul 01 2003

a(n) is the permanent of the n X n matrix M with M(i,j) = 1 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Dec 15 2003

Given n objects of distinct sizes (e.g. areas, volumes) such that each object is sufficiently large simultaneously to contain all previous objects, then n! is the total number of essentially different arrangements using all n objects. Arbitrary levels of nesting of objects is permitted within arrangements. (...sequence inspired by considering left-over moving boxes.). If the restriction exists that each object is only 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 (rshepherd2(AT)hotmail.com), Jan 14 2004

Stirling transform of a(n)=[2,2,6,24,120,...] is A052856(n)=[2,2,4,14,76,...]. - Michael Somos Mar 04 2004

Stirling transform of a(n)=[1,2,6,24,120,...] is A000670(n)=[1,3,13,75,...]. - Michael Somos Mar 04 2004

Stirling transform of a(n)=[0,2,6,24,120,...] is A052875(n)=[0,2,12,74,...]. - Michael Somos Mar 04 2004

Stirling transform of a(n-1)=[1,1,2,6,24,...] is A000629(n-1)=[1,2,6,26,...]. - Michael Somos Mar 04 2004

Stirling transform of a(n-1)=[0,1,2,6,24,...] is A002050(n-1)=[0,1,5,25,140,...]. - Michael Somos Mar 04 2004

Stirling transform of A006252(n)=[1,1,2,4,14,38,216,...] is a(n)=[1,2,6,24,120,...]. - Michael Somos Mar 04 2004

Stirling transform of -(-1)^n*A089064(n)=[1,0,1,-1,8,-26,194,...] is a(n-1)=[1,1,2,6,24,120,...]. - Michael Somos Mar 04 2004

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[e(n,k)s(k), k=0...n], where e(n,k) is a first-order Eulerian number [A008292]. - Ross La Haye (rlahaye(AT)new.rr.com), Feb 13 2005

1, 6, 120 are the only numbers which are both triangular and factorial. - Christopher M. Tomaszewski (cmt1288(AT)comcast.net), Mar 30 2005

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

a(n+1)=(n+1)!=1,2,6,.. has e.g.f. 1/(1-x)^2. - Paul Barry (pbarry(AT)wit.ie), Apr 22 2005

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 (amarnath_murthy(AT)yahoo.com), Jul 10 2005

The number of chains of maximal length in the power set of {1,2,...,n} ordered by the subset relation. - Rick L. Shepherd (rshepherd2(AT)hotmail.com), Feb 05 2006

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

a(n)=number of deco polyominoes of height n (n>=1; see definitions in the Barcucci et al. references). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 07 2006

a(n) = number of partition tableaux of size n. See Steingrimsson/Williams link for the definition. - David Callan (callan(AT)stat.wisc.edu), Oct 06 2006

Consider the n! permutation of the integer sequence [n]=1,2,...,n. The i-th permutation consists of ncycle(i) permutation cycles. Then, if the sum 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 (thomas.wieder(AT)t-online.de), Oct 11 2006

a(n) = 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 (callan(AT)stat.wisc.edu), Mar 30 2007

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 (rshepherd2(AT)hotmail.com), Jan 14 2004. - Thomas Wieder (thomas.wieder(AT)t-online.de), Nov 27 2007

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 (paulmuljadi(AT)yahoo.com), Apr 15 2008

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.

E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.

E. Barcucci, A. Del Lungo, R. Pinzani and R. Sprugnoli, La hauteur des polyominos dirige's verticalement convexes, Actes du 31e Se'minaire Lotharingien de Combinatoire, Publ. IRMA, Universite' Strasbourg I (1993).

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.

G. Labelle et al., Stirling numbers interpolation using permutations with forbidden subsequences, Discrete Math. 246 (2002), 177-195.

R. Ondrejka, 1273 exact factorials, Math. Comp., 24 (1970), 231.

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.

R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).

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.

R. P. Stanley, Recent Progress in Algebraic Combinatorics, Bull. Amer. Math. Soc., 40 (2003), 55-68.

D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp. 102 Penguin Books 1987.

R. W. Whitty, Rook polynomials on two-dimensional surfaces..., Discrete Math., 308 (2008), 674-683.

N. J. A. Sloane, The first 100 factorials: Table of n, n! for n = 0..100

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, December 1972 [alternative scanned copy].

David Applegate and N. J. A. Sloane, Table giving cycle index of S_0 through S_40 in Maple format [gzipped]

H. Bottomley, Illustration of initial terms

D. Butler, Factorials!

David Callan, Counting Stabilized-Interval-Free Permutations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.1.8.

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

R. M. Dickau, Permutation diagrams

H. Fripertinger, The elements of the symmetric group

H. Fripertinger, The elements of the symmetric group in cycle notation

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 20

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 297

C. Kimberling, Matrix Transformations of Integer Sequences, J. Integer Seqs., Vol. 6, 2003.

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

Paul Leyland, Generalized Cullen and Woodall numbers

N. E. Noerlund, Vorlesungen ueber Differenzenrechnung Springer 1924, p. 98.

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

F. Richman, Multiple precision arithmetic(Computing factorials up to 765!)

R. P. Stanley, A combinatorial miscellany

Einar Steingrimsson and Lauren K. Williams,Permutation tableaux and permutation patterns

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A. Walker, Factors of n!+-1

Sage Weil, The First 999 Factorials

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Laguerre Polynomial

Eric Weisstein's World of Mathematics, Diagonal Matrix

Wikipedia, Factorial

Index entries for sequences related to factorial numbers

Index entries for "core" sequences

a(0)=1; a(n)=n*a(n-1), n >= 1. n! ~ sqrt(2*Pi) * n^(n+1/2) / e^n (Stirling's approximation).

a(0)=1, a(n)=subs(x=1, diff(1/(2-x), x$n)), n=1, 2... - Karol A. Penson (penson(AT)lptl.jussieu.fr), Nov 12 2001

E.g.f.: 1/(1-x).

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)} . - DELEHAM Philippe, Jul 08 2004

Binomial transform of A000166. - Ross La Haye (rlahaye(AT)new.rr.com), Sep 21 2004

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 4! = (6+8+12+24) - (2+3+4+6+8+12) + 10 - 1 4! = 50 - 35 + 10 - 1 = 24 - Jon Perry (perry(AT)globalnet.co.uk), Nov 14 2005

a(0)=1, a(1)=1; a(n)=(n-1)*(a(n-1)+a(n-2)), n >= 2. - Matthew J. White (mattjameswhite(AT)hotmail.com), Feb 21 2006

a(n) = 1/Det[Table[(i+j)!/i!/(j+1)!,{i,1,n},{j,1,n}]] for n>0. This is a matrix with Catalan numbers on diagonal. - Alexander Adamchuk (alex(AT)kolmogorov.com), Jul 04 2006

Hankel transform of A074664 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 21 2007

There are 3! = 1*2*3 = 6 ways to arrange 3 letters {a,b,c}, namely abc, acb, bac, bca, cab, cba.

A000142 := n->n!; [ seq(n!, n=0..20) ];

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

(Maple program for computing cycle indices of symmetric groups)

M:=40: f:=array(0..M): f[0]:=1: lprint("n= ", 0); lprint(f[0]); f[1]:=x[1]: lprint("n= ", 1); lprint(f[1]);

for n from 2 to M do f[n]:=expand((1/n)*add( x[l]*f[n-l], l=1..n)); lprint("n= ", n); lprint(f[n]); od:

with(combinat):seq((stirling1(j+1, 1)*(stirling2(j+1, 1))*(-1)^j), j=0..20); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 30 2007

with(combstruct):ZL0:=[S, {S=Set(Cycle(Z, card>0))}, labeled]: seq(count(ZL0, size=n), n=0..20); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Sep 26 2007

a[n_] := n!; Table[a[n], {n, 0, 20}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Mar 30 2006

(AXIOM) [factorial(n) for n in 0..10]

(MAGMA) a:= func< n | Factorial(n) >; [ a(n) : n in [0..10]];

(PARI) a(n)=if(n<0, 0, n!)

Cf. A047920, A048631, A003422, A000165, A001563, A001044, A010050, A009445, A038507, A033312.

Cf. A034886.

Factorial base representation: A007623.

Adjacent sequences: A000139 A000140 A000141 this_sequence A000143 A000144 A000145

Sequence in context: A062919 A072133 A072167 this_sequence A104150 A124355 A133942

Cf. A012245.

core,easy,nonn,nice,new

njas