Search: id:A000165 Results 1-1 of 1 results found. %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 ii, 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, Table of n, a(n) for n=0..100 %H A000165 P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5. %H A000165 Hamed Hatami, Pooya Hatami, Perfect dominating sets in the Cartesian products of prime cycles. %H A000165 Jason D. Hildebrand, Differentiating Arctan(x) %H A000165 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 136 %H A000165 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 %H A000165 M. Z. Spivey and L. L. Steil, The k-Binomial Transforms and the Hankel Transform, J. Integ. Seqs. Vol. 9 (2006), #06.1.1. %H A000165 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A000165 Eric Weisstein's World of Mathematics, Graph Automorphism %H A000165 Index entries for sequences related to factorial numbers %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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