%I A000165 M1878 N0742
%S A000165 1,2,8,48,384,3840,46080,645120,10321920,185794560,3715891200,
%T A000165 81749606400,1961990553600,51011754393600,1428329123020800,
%U A000165 42849873690624000,1371195958099968000,46620662575398912000
%N A000165 Double factorial numbers: (2n)!! = 2^n*n!.
%C A000165 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
%C A000165 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
%C A000165 Appears to be the BinomialMean transform of A001907. See A075271. - John
W. Layman (layman(AT)math.vt.edu), Sep 28 2002
%C A000165 Number of n X n monomial matrices with entries 0, +-1.
%C A000165 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
%C A000165 Also number of linear signed orders.
%C A000165 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
%C A000165 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
%C A000165 1 + (1/2)x - (1/8)x^2 - (1/48)x^3 + (1/384)x^4 +... = sqrt(1+sin(x)).
%C A000165 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)gmail.com),
Jan 14 2006
%C A000165 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
%C A000165 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
%C A000165 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 (jvospost3(AT)gmail.com), Jan 03 2007
%C A000165 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
%C A000165 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
%C A000165 Row sums of A028338. [From Paul Barry (pbarry(AT)wit.ie), Feb 07 2009]
%C A000165 a(n) is the number of ways to seat n married couples in a row so that
everyone is next to their spouse. Compare A007060 [From Geoffrey
Critzer (critzer.geoffrey(AT)usd443.org), Mar 29 2009]
%C A000165 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 21 2009:
(Start)
%C A000165 Equals (-1)^n * (1, 1, 2, 8, 48,...) dot (1, -3, 5, -7, 9,...).
%C A000165 Example: a(4) = 384 = (1, 1, 2, 8, 48) dot (1, -3, 5, -7, 9) = (1, -3,
10, -56, 432). (End)
%C A000165 exp(x/2)=sum(n>=0,x^n/a(n)) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com),
Sep 07 2009]
%D A000165 Peter C. Fishburn, Signed Orders, Choice Probabilities and Linear Polytopes,
Journal of Mathematical Psychology, Volume 45, Issue 1, (2001), pp.
53-80.
%D A000165 G. Gordon, The answer is 2^n*n! What is the question? Amer. Math. Monthly,
106 (1999), 636-645.
%D A000165 McDonnell, Eugene, "Magic Squares and Permutations", APL Quote Quad 7.3
(Fall 1976)
%D A000165 B. E. Meserve, Double factorials, Amer. Math. Monthly, 55 (1948), 425-426.
%D A000165 R. Ondrejka, Tables of double factorials, Math. Comp., 24 (1970), 231.
%D A000165 R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial
Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
%D A000165 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A000165 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A000165 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 A000165 T. D. Noe, <a href="b000165.txt">Table of n, a(n) for n=0..100</a>
%H A000165 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
Sequences realized by oligomorphic permutation groups</a>, J. Integ.
Seqs. Vol. 3 (2000), #00.1.5.
%H A000165 Hamed Hatami, Pooya Hatami, <a href="http://arXiv.org/list/math.CO/0701018.pdf">
Perfect dominating sets in the Cartesian products of prime cycles</
a>.
%H A000165 Jason D. Hildebrand, <a href="http://www.opensky.ca/~jdhildeb/arctan/
arctan_diff.html">Differentiating Arctan(x)</a>
%H A000165 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=136">
Encyclopedia of Combinatorial Structures 136</a>
%H A000165 Alexsandar Petojevic, <a href="http://www.cs.uwaterloo.ca/journals/JIS/
index.html">The Function vM_m(s; a; z) and Some Well-Known Sequences</
a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7
%H A000165 M. Z. Spivey and L. L. Steil, <a href="http://www.cs.uwaterloo.ca/journals/
JIS/VOL9/Spivey/spivey7.pdf">The k-Binomial Transforms and the Hankel
Transform</a>, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.
%H A000165 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
DoubleFactorial.html">Link to a section of The World of Mathematics.</
a>
%H A000165 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
GraphAutomorphism.html">Graph Automorphism</a>
%H A000165 <a href="Sindx_Fa.html#factorial">Index entries for sequences related
to factorial numbers</a>
%F A000165 E.g.f.: 1/(1-2*x).
%F A000165 a(n)=2n*a(n-1), n>0, a(0)=1. - Paul Barry (pbarry(AT)wit.ie), Aug 26
2004
%F A000165 This is the binomial mean transform of A001907. See Spivey and Steil
(2006). - Michael Z. Spivey (mspivey(AT)ups.edu), Feb 26 2006
%F A000165 a(n)=int(x^n*exp(-x/2)/2,x,0,infty); - Paul Barry (pbarry(AT)wit.ie),
Jan 28 2008
%F A000165 Let b(n)=b(n-1)+2; then a(n)=b(n)*a(n-1). - Roger L. Bagula (rlbagulatftn(AT)yahoo.com),
Sep 17 2008
%F A000165 G.f.: 1/(1-2x/(1-2x/(1-4x/(1-4x/(1-6x/(1-6x/(1-.... (continued fraction).
[From Paul Barry (pbarry(AT)wit.ie), Feb 07 2009]
%F A000165 a(n) = 2^n*n! i.e.(2= 1*2, 8 = 2*4, 48 = 6*8, 384 =24*16, 3840= 120*32...)
[From Gary Detlefs (gdetlefs(AT)aol.com), Aug 09 2009]
%F A000165 a(n)=A006882(2*n). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl),
Oct 20 2009]
%e A000165 The following permutations and their reversals are all of the permutations
of order 5 having the double-downgrade property:
%e A000165 0 1 2 3 4
%e A000165 0 3 2 1 4
%e A000165 1 0 2 4 3
%e A000165 1 4 2 0 3
%p A000165 A000165 := proc(n) option remember; if n <= 1 then 1 else n*A000165(n-2);
fi; end;
%p A000165 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
%p A000165 restart: G(x):=(1-2*x)^(-1): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1],
x) od: x:=0: seq(f[n],n=0..17);# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Apr 03 2009]
%p A000165 A000165 := proc(n) doublefactorial(2*n) ; end proc; seq(A000165(n),n=0..10)
; [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 20 2009]
%t A000165 a[0] = 1; a[p_] :=2*p*a[p - 1] ; a /@ Range[0, 19] - Zerinvary Lajos
(zerinvarylajos(AT)yahoo.com), Mar 29 2007
%t A000165 k = 2; b[1] = 2; b[n_] := b[n] = b[n - 1] + k; a[0] = 1; a[1] = 2; a[n_]
:= a[n] = a[n - 1]*b[n]; Table[a[n], {n, 0, 20}] - Roger L. Bagula
(rlbagulatftn(AT)yahoo.com), Sep 17 2008
%t A000165 a[n_]:=(2*n)!!; [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec
13 2008]
%Y A000165 Cf. A006882, A000142 (n!), A001147 ((2n-1)!!), A010050, A002454, A039683.
%Y A000165 Cf. A008544, A001813, A047055, A047657, A084947, A084948, A084949.
%Y A000165 Cf. A001813.
%Y A000165 This sequence gives the row sums in A060187.
%Y A000165 Sequence in context: A003576 A095989 A124453 this_sequence A109664 A009812
A063075
%Y A000165 Adjacent sequences: A000162 A000163 A000164 this_sequence A000166 A000167
A000168
%K A000165 nonn,easy,nice
%O A000165 0,2
%A A000165 N. J. A. Sloane (njas(AT)research.att.com).
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