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

Number of terms in a determinant when writing down all multiplication permutations. [From Mats O. Granvik (mgranvik(AT)abo.fi), Sep 12 2008]

Triangle A144107 has row sums = n!(n>0) with right border n! and left border A003319, the INVERTi transform of (1, 2, 6, 24,...) [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 11 2008]

Equals INVERT transform of A052186: (1, 0, 1, 3, 14, 77,...) and row sums of triangle A144108. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 11 2008]

Contribution from A. Umar (aumarh(AT)squ.edu.om), Oct 12 2008: (Start)

a(n) is also the number of order-decreasing full transformations (of an n-chain).

a(n-1) is also the number of nilpotent order-decreasing full transformations (of an n-chain). (End)

Contribution from Calin D. Morosan (cd_moros(AT)alumni.concordia.ca), Nov 28 2008: (Start)

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). (End)

Sum_{n>=0} 1/a(n) = e [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Mar 03 2009]

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 ). [From Kailasam Viswanathan Iyer (kvi(AT)nitt.edu), Mar 18 2009]

Chromatic invariant of the sun graph S_{n-2}

It appears that a(n+1) is the inverse binomial transform of A000255. [From Timothy Hopper (timothyhopper(AT)hotmail.co.uk), Aug 20 2009]

a(n) is also the determinant of an square matrix, An, whose coefficients are the reciprocals of beta function: a{i,j}=1/beta(i,j), det(An)=n! [From Barbarel Tres Mil (barbarel3000(AT)yahoo.es), Sep 21 2009]

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 20 2009: (Start)

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.

(End)

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.

S.B.Akers and B.Krishnamurthy, "A group-theoretic model for symmetric interconnection networks" , IEEE Trans. Comput., 38(4), April 1989, 555-566. [From Kailasam Viswanathan Iyer (kvi(AT)nitt.edu), Mar 18 2009]

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.

M. Bhargava, The Factorial Function and Generalizations, Amer. Math. Monthly 107 (2000) 783-799. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jul 26 2009]

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).

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).

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.

Umar, A. On the semigroups of order-decreasing finite full transformations, Proc. Roy. Soc. Edinburgh 120A (1992), 129-142. [From A. Umar (aumarh(AT)squ.edu.om), Oct 12 2008]

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

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].

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

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 18

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

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

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

G. Villemin's Almanach of Numbers, Factorielles

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

Eric Weisstein's World of Mathematics, Chromatic Invariant

Wikipedia, Factorial

Index entries for sequences related to factorial numbers

Index entries for "core" sequences

Barbarel Tres Mil, Beta function matrix determinant Psychedelic Geometry blogspot-09/21/09 [From Barbarel Tres Mil (barbarel3000(AT)yahoo.es), Sep 21 2009]

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

For n>=2, a(n-2)=(-1)^n*sum((j+1)*stirling1(n,j+1),j=0..n-1); [From Milan R. Janjic (agnus(AT)blic.net), Dec 14 2008]

Contribution from Paul Barry (pbarry(AT)wit.ie), Jan 15 2009: (Start)

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.

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)

A007814(a(n))=n-A000120(n). [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Jul 20 2009]

a(n) = Prod_{p prime} p^{Sum_{k>0} [n/p^k]} by Legendre's formula for the highest power of a prime dividing n!. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jul 24 2009]

a(n) = A053657(n)/A163176(n) for n > 0. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jul 26 2009]

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. [From Timothy Hopper (timothyhopper(AT)hotmail.co.uk), Aug 12 2009]

a(0)=1, a(1)=1; a(n)=(a(n-1)^2+a(n-1)*a(n-2))/a(n-2), n>=2 [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Sep 21 2009]

a(n)=Gamma(n+1) [From Barbarel Tres Mil (barbarel3000(AT)yahoo.es), Sep 21 2009]

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->mul(numer (k/(k+1)), k=1..n): seq(a(n), n=0..20); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 26 2008

a:=n->mul(denom (1/(k+2)), k=0..n): seq(a(n), n=-2..18); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 26 2008

restart:a:= proc(n) option remember; if n=0 then 1 else add(a(n-1), j=0..n-1) fi end: seq (a(n), n=0..20); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 29 2009]

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

Table[(-1)^(n + 1)* Sum[(-1)^(n - k) k (-1)^(n - k) StirlingS1[n + 1, k + 1], {k, 0, n}], {n, 1, 21}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 08 2009]

a[n_] := Product[Prime[j]^Sum[Floor[n/Prime[j]^k], {k, 1, Ceiling[Log[n]/Log[Prime[j]]]}], {j, 1, n}]; Table[a[n], {n, 1, 20}] [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jul 24 2009]

(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!)

(Other) sage: [stirling_number1(n, 1) for n in xrange(1, 22)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 16 2009]

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

Cf. A034886.

Factorial base representation: A007623.

Cf. A012245.

Cf. A144108, A052186, A144107, A003319 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 11 2008]

Complement of A063992. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 11 2008]

Row products of A139547. [From Mats Granvik (mats.granvik(AT)abo.fi), Jun 28 2009]

Cf. A053657, A163176. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jul 26 2009]

Sequence in context: A072167 A154659 A155456 this_sequence A104150 A124355 A133942

Adjacent sequences: A000139 A000140 A000141 this_sequence A000143 A000144 A000145

core,easy,nonn,nice

N. J. A. Sloane (njas(AT)research.att.com).