0,4

For n >= 3, a(n-1) is also the number of ways that a 3-cycle in the symmetric group S_n can be written as a product of 2 long cycles (of length n). - Ahmed Fares (ahmedfares(AT)my-deja.com), Aug 14 2001

a(n) is the number of Hamiltonian circuit masks for an n X n adjacency matrix of an undirected graph. - Chad R. Brewbaker (crb002(AT)iastate.edu), Jan 31 2003

a(n) is the number of necklaces one can make with n distinct beads: n! bead permutations, divide by two to represent flipping the necklace over, divide by n to represent rotating the necklace. Related to Stirling numbers of the first kind, Stirling cycles. - Chad R. Brewbaker (crb002(AT)iastate.edu), Jan 31 2003

Number of increasing runs in all permutations of [n-1] (n>=2). Example: a(4)=12 because we have 12 increasing runs in all the permutations of [3] (shown in parentheses): (123), (13)(2), (3)(12), (2)(13), (23)(1), (3)(2)(1). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 28 2004

Minimum permanent over all n X n (0,1)-matrices with exactly n/2 zeros. - Simone Severini (ss54(AT)york.ac.uk), Oct 15 2004

Comment from John Perry, Sep 20 2008: The number of permutations of 1..n that have 2 following 1 for n >= 1 is 0,1,3,12,60,360,2520,20160,... .

Starting (1, 3, 12, 60,...) = binomial transform of A000153: (1, 2, 7, 32, 181,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 25 2008]

First column of A092582. [From Mats Granvik (mats.granvik(AT)abo.fi), Feb 08 2009]

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

The asymptotic expansion of the higher order exponential integral E(x,m=1,n=3) ~ exp(-x)/x*(1 - 3/x + 12/x^2 - 60/x^3 + 360/x^4 - 2520/x^5 + 20160/x^6 - 81440/x^7 + ...) leads to the sequence given above. See A163931 and A130534 for more information.

(End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Mitrinovic, D. S.; Mitrinovic, R. S.; Tableaux d'une classe de nombres relies aux nombres de Stirling. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 88.

S-Z Song, S-G Hwang, S-H Rim, G-S Cheon, Extremes of permanents of (0,1)-matrices. Special issue on the Combinatorial Matrix Theory Conference (Pohang, 2002). Linear Algebra Appl. 373 (2003), 197-210.

N. J. A. Sloane, Table of n, a(n) for n = 0..100

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

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 262

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

Xah Lee, Combinatorics: Loop in n points

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

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, Even Permutation

Eric Weisstein's World of Mathematics, Odd Permutation

Index entries for sequences related to factorial numbers

a(n) = numerator(n!/2) and A141044(n) = denominator(n!/2).

a(0) = a(1) = a(2) = 1; a(n)=n*a(n-1) for n>3. - Chad R. Brewbaker (crb002(AT)iastate.edu), Jan 31 2003 [Corrected by N. J. A. Sloane (njas(AT)research.att.com), Jul 25 2008]

a(0) = 0, a(1) = 1; a(n) = sum k*a(k) for k = 1 to n-1. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Oct 29 2002

Stirling transform of a(n+1)=[1, 3, 12, 160, ...] is A083410(n)=[1, 4, 22, 154, ...]. - Michael Somos Mar 04 2004

First Eulerian transform of A000027. See A000142 for definition of FET. - Ross La Haye (rlahaye(AT)new.rr.com), Feb 14 2005

a(n)=sum{k=0..n, (-1)^(n-k-1)T(n-1, k)cos(pi(n-k-1)/2)^2}+0^n; T(n, k)=abs(A008276(n, k)). - Paul Barry (pbarry(AT)wit.ie), Apr 18 2005

E.g.f.: (2-x^2)/(2-2*x). E.g.f. of a(n+2),n>=0, is 1/(1-x)^3.

a(n+1)= A136656(n,1)*(-1)^n, n>=1.

seq(mul((k), k=3..n), n=0..20); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Sep 14 2007

a[ -1]:=1:a[0]:=1:a[1]:=1:for n from 2 to 50 do a[n]:=(a[n-1]*(n+1)^2) od: seq(sqrt(a[n]), n=-1..19); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 09 2008

with (combinat):seq(count(Partition((n!+1)), size=2), n=0..20); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 28 2008

f[n_]:=If[n>1, n, 1]; a=2; lst={1}; Do[a=n*a-a; AppendTo[lst, f[a/4]], {n, 2, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), May 28 2009]

(PARI) a(n)=if(n<2, n>=0, n!/2)

Cf. A000142, A049444, A049459. a(n+1)= A046089(n, 1), n >= 1 (first column of triangle).

A000153 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 25 2008]

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 18 2009: (Start)

Cf. A161739 (q(n) sequence).

(End)

Sequence in context: A062569 A089057 A077134 this_sequence A105752 A053532 A159867

Adjacent sequences: A001707 A001708 A001709 this_sequence A001711 A001712 A001713

nonn,easy,nice

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

More terms from Larry Reeves (larryr(AT)acm.org), Aug 20 2001

Further from Simone Severini (ss54(AT)york.ac.uk), Oct 15 2004