0,2

a(n) is also the size of automorphism group of the graph (edge graph) of the n dimensional hypercube and also of the geometric automorphism group of the hypercube (the two groups are isomorphic). This group is an extension of an elementary abelian group (C_2)^n by S_n. (C_2 is the cyclic group with two elements and S_n is the symmetric group) - Avi Peretz (njk(AT)netvision.net.il), Feb 21 2001

Then a(n) appears in the power series: sqrt(1+sin(y))=sum(n>=0,(-1)^floor(n/2)*y^(n)/a(n)) and sqrt((1+cos(y))/2)=sum(n>=0,(-1)^n*y^(2n)/a(2n)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Feb 02 2002

Appears to be the BinomialMean transform of A001907. See A075271. - John W. Layman (layman(AT)math.vt.edu), Sep 28 2002

Number of n X n monomial matrices with entries 0, +-1.

a(n) = A001044(n)/A000142(n)*A000079(n) = product(2*i+2,i=0..n-1) = 2^n*pochhammer(1,n) - Daniel Dockery (peritus(AT)gmail.com) Jun 13, 2003

Also number of linear signed orders.

Define a "downgrade" to be the permutation d which places the items of a permutation p in descending order. This note concerns those permutations that are equal to their double-downgrades. The number of permutations of order 2n having this property are equinumerous with those of order 2n+1. a(n) = number of double-downgrading permutations of order 2n and 2n+1. - Eugene McDonnell (eemcd(AT)mac.com), Oct 27 2003

a(n)=(integral_{x=0 to pi/2} cos(x)^(2*n+1) dx) where the denominators are b(n)= (2*n)!/(n!*2^n). - Al Hakanson (hawkuu(AT)excite.com), Mar 02 2004

1 + (1/2)x - (1/8)x^2 - (1/48)x^3 + (1/384)x^4 +... = sqrt(1+sin(x)).

a(n)*(-1)^n = coefficient of the leading term of the (n+1)-th derivative of arctan(x), see Hildebrand link. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jan 14 2006

a(n) is the Pfaffian of the skew-symmetric 2n X 2n matrix whose (i,j) entry is j for i<j. - David Callan (callan(AT)stat.wisc.edu), Sep 25 2006

a(n) is the number of increasing plane trees with n+1 edges. (In a plane tree, each subtree of the root is an ordered tree but the subtrees of the root may be cyclically rotated.) Increasing means the vertices are labeled 0,1,2,...,n+1 and each child has a greater label than its parent. Cf. A001147 for increasing ordered trees, A000142 for increasing unordered trees, and A000111 for increasing 0-1-2 trees. - David Callan (callan(AT)stat.wisc.edu), Dec 22 2006

Hamed Hatami and Pooya Hatami prove that this is an upper bound on the cardinality of any minimal dominating set in C_{2n+1}^n, the Cartesian product of n copies of the cycle of size 2n+1, where 2n+1 is a prime. - Jonathan Vos Post (jvospost2(AT)yahoo.com), Jan 03 2007

This sequence and (1,-2,0,0,0,0,...) form a reciprocal pair under the list partition transform and associated operations described in A133314. - Tom Copeland (tcjpn(AT)msn.com), Oct 29 2007

a(n) = number of permutations of the multiset {1,1,2,2,...,n,n,n+1,n+1} such that between the two occurrences of i, there is exactly one entry >i, for i=1,2,...,n. Example: a(2) = 8 counts 121323, 131232, 213123, 231213, 232131, 312132, 321312, 323121. Proof. There is always exactly one entry between the two 1s (when n>=1). Given a permutation p in A(n) (counted by a(n)), record the position i of the first 1, then delete both 1s and subtract 1 from every entry to get a permutation q in A(n-1). The mapping p -> (i,q) is a bijection from A(n) to the Cartesian product [1,2n] X A(n-1). - David Callan (callan(AT)stat.wisc.edu), Nov 29 2007

G. Gordon, The answer is 2^n*n! What is the question? Amer. Math. Monthly, 106 (1999), 636-645.

B. E. Meserve, Double factorials, Amer. Math. Monthly, 55 (1948), 425-426.

R. Ondrejka, Tables of double factorials, Math. Comp., 24 (1970), 231.

Peter C. Fishburn, Signed Orders, Choice Probabilities and Linear Polytopes, Journal of Mathematical Psychology, Volume 45, Issue 1, (2001), pp. 53-80.

McDonnell, Eugene, "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).

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.

T. D. Noe, 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.

Jason D. Hildebrand, Differentiating Arctan(x)

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 136

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

M. Z. Spivey and L. L. Steil, The k-Binomial Transforms and the Hankel Transform, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.

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

Eric Weisstein's World of Mathematics, Graph Automorphism

Index entries for sequences related to factorial numbers

Hamed Hatami, Pooya Hatami, Perfect dominating sets in the Cartesian products of prime cycles.

E.g.f.: 1/(1-2*x).

a(n)=2n*a(n-1), n>0, a(0)=1. - Paul Barry (pbarry(AT)wit.ie), Aug 26 2004

This is the binomial mean transform of A001907. See Spivey and Steil (2006). - Michael Z. Spivey (mspivey(AT)ups.edu), Feb 26 2006

a(n)=int(x^n*exp(-x/2)/2,x,0,infty); - Paul Barry (pbarry(AT)wit.ie), Jan 28 2008

The following permutations and their reversals are all of the permutations of order 5 having the double-downgrade property:

0 1 2 3 4

0 3 2 1 4

1 0 2 4 3

1 4 2 0 3

A000165 := proc(n) option remember; if n <= 1 then 1 else n*A000165(n-2); fi; end;

ZL:=[S, {a = Atom, b = Atom, S = Prod(X, Sequence(Prod(X, b))), X = Sequence(b, card >= 0)}, labelled]: seq(combstruct[count](ZL, size=n), n=0..17); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 26 2008

a[0] = 1; a[p_] :=2*p*a[p - 1] ; a /@ Range[0, 19] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 29 2007

Cf. A006882, A000142 (n!), A001147 ((2n-1)!!), A010050, A002454, A039683.

Cf. A008544, A001813, A047055, A047657, A084947, A084948, A084949.

Cf. A001813.

Adjacent sequences: A000162 A000163 A000164 this_sequence A000166 A000167 A000168

Sequence in context: A003576 A095989 A124453 this_sequence A109664 A009812 A063075

nonn,easy,nice

njas