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%I M1878 N0742 #320 Jul 23 2023 08:49:47
%S 1,2,8,48,384,3840,46080,645120,10321920,185794560,3715891200,
%T 81749606400,1961990553600,51011754393600,1428329123020800,
%U 42849873690624000,1371195958099968000,46620662575398912000,1678343852714360832000,63777066403145711616000
%N Double factorial of even numbers: (2n)!! = 2^n*n!.
%C a(n) is also the size of the 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 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_, Feb 02 2002
%C Appears to be the BinomialMean transform of A001907. See A075271. - _John W. Layman_, Sep 28 2002
%C Number of n X n monomial matrices with entries 0, +-1.
%C a(n) = A001044(n)/A000142(n)*A000079(n) = Product_{i=0..n-1} (2*i+2) = 2^n*Pochhammer(1,n). - Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003
%C Also number of linear signed orders.
%C 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 a(n) = (Integral_{x=0..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 1 + (1/2)x - (1/8)x^2 - (1/48)x^3 + (1/384)x^4 + ... = sqrt(1+sin(x)).
%C a(n)*(-1)^n = coefficient of the leading term of the (n+1)-th derivative of arctan(x), see Hildebrand link. - _Reinhard Zumkeller_, Jan 14 2006
%C a(n) is the Pfaffian of the skew-symmetric 2n X 2n matrix whose (i,j) entry is j for i<j. - _David Callan_, Sep 25 2006
%C 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_, Dec 22 2006
%C 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_, Jan 03 2007
%C 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_, Oct 29 2007
%C 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_, Nov 29 2007
%C Row sums of A028338. - _Paul Barry_, Feb 07 2009
%C 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. - _Geoffrey Critzer_, Mar 29 2009
%C From _Gary W. Adamson_, Apr 21 2009: (Start)
%C Equals (-1)^n * (1, 1, 2, 8, 48, ...) dot (1, -3, 5, -7, 9, ...).
%C Example: a(4) = 384 = (1, 1, 2, 8, 48) dot (1, -3, 5, -7, 9) = (1, -3, 10, -56, 432). (End)
%C exp(x/2)=sum(n>=0,x^n/a(n)). - _Jaume Oliver Lafont_, Sep 07 2009
%C Assuming n starts at 0, a(n) appears to be the number of Gray codes on n bits. It certainly is the number of Gray codes on n bits isomorphic to the canonical one. Proof: There are 2^n different starting positions for each code. Also, each code has a particular pattern of bit positions that are flipped (for instance, 1 2 1 3 1 2 1 for n=3), and these bit position patterns can be permuted in n! ways. - D. J. Schreffler (ds1404(AT)txstate.edu), Jul 18 2010
%C E.g.f. of 0,1,2,8,... is x/(1-2x/(2-2x/(3-8x/(4-8x/(5-18x/(6-18x/(7-... (continued fraction). - _Paul Barry_, Jan 17 2011
%C Number of increasing 2-colored trees with choice of two colors for each edge. In general, if we replace 2 with k we get the number of increasing k-colored trees. For example, for k=3 we get the triple factorial numbers. - _Wenjin Woan_, May 31 2011
%C a(n) = row sums of triangle A193229. - _Gary W. Adamson_, Jul 18 2011
%C Also the number of permutations of 2n (or of 2n+1) that are equal to their reverse-complements. (See the Egge reference.) Note that the double-downgrade described in the preceding comment (McDonnell) is equivalent to the reverse-complement. - _Justin M. Troyka_, Aug 11 2011
%C The e.g.f. can be used to form a generator, [1/(1-2x)] d/dx, for A000108, so a(n) can be applied to A145271 to generate the Catalan numbers. - _Tom Copeland_, Oct 01 2011
%C The e.g.f. of 1/a(n) is BesselI(0,sqrt(2*x)). See Abramowitz-Stegun (reference and link under A008277), p. 375, 9.6.10. - _Wolfdieter Lang_, Jan 09 2012
%C a(n) = order of the largest imprimitive group of degree 2n with n systems of imprimitivity (see [Miller], p. 203). - _L. Edson Jeffery_, Feb 05 2012
%C Row sums of triangle A208057. - _Gary W. Adamson_, Feb 22 2012
%C a(n) is the number of ways to designate a subset of elements in each n-permutation. a(n) = A000142(n) + A001563(n) + A001804(n) + A001805(n) + A001806(n) + A001807(n) + A035038(n) * n!. - _Geoffrey Critzer_, Nov 08 2012
%C For n>1, a(n) is the order of the Coxeter groups (also called Weyl groups) of types B_n and C_n. - _Tom Edgar_, Nov 05 2013
%C For m>0, k*a(m-1) is the m-th cumulant of the chi-squared probability distribution for k degrees of freedom. - _Stanislav Sykora_, Jun 27 2014
%C a(n) with 0 prepended is the binomial transform of A120765. - _Vladimir Reshetnikov_, Oct 28 2015
%C Exponential self-convolution of A001147. - _Vladimir Reshetnikov_, Oct 08 2016
%C Also the order of the automorphism group of the n-ladder rung graph. - _Eric W. Weisstein_, Jul 22 2017
%C a(n) is the order of the group O_n(Z) = {A in M_n(Z): A*A^T = I_n}, the group of n X n orthogonal matrices over the integers. - _Jianing Song_, Mar 29 2021
%C a(n) is the number of ways to tile a (3n,3n)-benzel or a (3n+1,3n+2)-benzel using left stones and two kinds of bones; see Defant et al., below. - _James Propp_, Jul 22 2023
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A000165/b000165.txt">Table of n, a(n) for n = 0..100</a>
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Barry4/barry271.html">General Eulerian Polynomials as Moments Using Exponential Riordan Arrays</a>, Journal of Integer Sequences, 16 (2013), #13.9.6.
%H Paul Barry, <a href="https://arxiv.org/abs/1803.10297">Generalized Eulerian Triangles and Some Special Production Matrices</a>, arXiv:1803.10297 [math.CO], 2018.
%H Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, <a href="https://www.emis.de/journals/JIS/VOL21/Falcao/falcao2.html">Combinatorial Identities Associated with a Multidimensional Polynomial Sequence</a>, J. Int. Seq., Vol. 21 (2018), Article 18.7.4.
%H P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H CombOS - Combinatorial Object Server, <a href="http://combos.org/cperm">Generate colored permutations</a>
%H R. Coquereaux and J.-B. Zuber, <a href="https://arxiv.org/abs/1507.03163">Maps, immersions and permutations</a>, arXiv preprint arXiv:1507.03163 [math.CO], 2015.
%H Colin Defant, Rupert Li, James Propp, and Benjamin Young, <a href="https://arxiv.org/abs/2209.05717">Tilings of Benzels via the Abacus Bijection</a>, arXiv preprint, arXiv:2209.05717 [math.CO], 2022.
%H Eric S. Egge, <a href="http://dx.doi.org/10.1007/s00026-007-0327-9">Restricted symmetric permutations</a>, Ann. Combin., 11 (2007), 405-434.
%H Peter C. Fishburn, <a href="http://dx.doi.org/10.1006/jmps.1999.1288">Signed Orders, Choice Probabilities and Linear Polytopes</a>, Journal of Mathematical Psychology, Volume 45, Issue 1, (2001), pp. 53-80.
%H Joël Gay and Vincent Pilaud, <a href="https://arxiv.org/abs/1804.06572">The weak order on Weyl posets</a>, arXiv:1804.06572 [math.CO], 2018.
%H G. Gordon, <a href="http://www.jstor.org/stable/2589493">The answer is 2^n*n! What is the question?</a>, Amer. Math. Monthly, 106 (1999), 636-645.
%H Guo-Niu Han, <a href="http://www-irma.u-strasbg.fr/~guoniu/papers/p77puzzle.pdf">Enumeration of Standard Puzzles</a>
%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a> [Cached copy]
%H Hamed Hatami and Pooya Hatami, <a href="https://arxiv.org/abs/math/0701018">Perfect dominating sets in the Cartesian products of prime cycles</a>, arXiv:math/0701018 [math.CO], 2006-2009.
%H Jason D. Hildebrand, <a href="http://www.opensky.ca/~jdhildeb/arctan/arctan_diff.html">Differentiating Arctan(x)</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=136">Encyclopedia of Combinatorial Structures 136</a>
%H L. C. Larson, <a href="/A000900/a000900_1.pdf">The number of essentially different nonattacking rook arrangements</a>, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181. [Annotated scan of pages 180 and 181 only]
%H E. Lucas, <a href="/A000899/a000899.pdf">Théorie des nombres</a> (annotated scans of a few selected pages)
%H Eugene McDonnell, <a href="http://www.jsoftware.com/papers/eem/magicsq.htm">Magic Squares and Permutations</a>, APL Quote Quad 7.3 (Fall 1976).
%H B. E. Meserve, <a href="http://www.jstor.org/stable/2306136">Double Factorials</a>, American Mathematical Monthly, 55 (1948), 425-426.
%H G. A. Miller, <a href="http://dx.doi.org/10.1090/S0002-9904-1918-03043-7">Groups formed by special matrices</a>, Bull. Amer. Math. Soc. 24 (1918), 203-206.
%H R. Ondrejka, <a href="http://dx.doi.org/10.1090/S0025-5718-70-99856-X">Tables of double factorials</a>, Math. Comp., 24 (1970), 231.
%H Alexsandar Petojevic, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL5/Petojevic/petojevic5.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 Luis Manuel Rivera, <a href="https://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
%H R. W. Robinson, <a href="http://dx.doi.org/10.1007/BFb0097382">Counting arrangements of bishops</a>, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
%H M. Z. Spivey and L. L. Steil, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Spivey/spivey7.html">The k-Binomial Transforms and the Hankel Transform</a>, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DoubleFactorial.html">Double Factorial</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphAutomorphism.html">Graph Automorphism</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LadderRungGraph.html">Ladder Rung Graph</a>
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>
%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F E.g.f.: 1/(1-2*x).
%F D-finite with recurrence a(n) = 2*n * a(n-1), n>0, a(0)=1. - _Paul Barry_, Aug 26 2004
%F This is the binomial mean transform of A001907. See Spivey and Steil (2006). - Michael Z. Spivey (mspivey(AT)ups.edu), Feb 26 2006
%F a(n) = Integral_{x>=0} x^n*exp(-x/2)/2 dx. - _Paul Barry_, Jan 28 2008
%F G.f.: 1/(1-2x/(1-2x/(1-4x/(1-4x/(1-6x/(1-6x/(1-.... (continued fraction). - _Paul Barry_, Feb 07 2009
%F a(n) = A006882(2*n). - _R. J. Mathar_, Oct 20 2009
%F From _Gary W. Adamson_, Jul 18 2011: (Start)
%F a(n) = upper left term in M^n, M = a production matrix (twice Pascal's triangle deleting the first "2", with the rest zeros; cf. A028326):
%F 2, 2, 0, 0, 0, 0, ...
%F 2, 4, 2, 0, 0, 0, ...
%F 2, 6, 6, 2, 0, 0, ...
%F 2, 8, 12, 8, 2, 0, ...
%F 2, 10, 20, 20, 10, 2, ...
%F ... (End)
%F From _Sergei N. Gladkovskii_, Apr 11 2013, May 01 2013, May 24 2013, Sep 30 2013, Oct 27 2013: (Start)
%F Continued fractions:
%F G.f.: 1 + x*(Q(0) - 1)/(x+1) where Q(k) = 1 + (2*k+2)/(1-x/(x+1/Q(k+1))).
%F G.f.: 1/Q(0) where Q(k) = 1 + 2*k*x - 2*x*(k+1)/Q(k+1).
%F G.f.: G(0)/2 where G(k) = 1 + 1/(1 - x*(2*k+2)/(x*(2*k+2) + 1/G(k+1))).
%F G.f.: 1/Q(0) where Q(k) = 1 - x*(4*k+2) - 4*x^2*(k+1)^2/Q(k+1).
%F G.f.: R(0) where R(k) = 1 - x*(2*k+2)/(x*(2*k+2)-1/(1-x*(2*k+2)/(x*(2*k+2) -1/R(k+1)))). (End)
%F a(n) = (2n-2)*a(n-2) + (2n-1)*a(n-1), n>1. - _Ivan N. Ianakiev_, Aug 06 2013
%F From _Peter Bala_, Feb 18 2015: (Start)
%F Recurrence equation: a(n) = (3*n - 1)*a(n-1) - 2*(n - 1)^2*a(n-2) with a(1) = 2 and a(2) = 8.
%F The sequence b(n) = A068102(n) also satisfies this second-order recurrence. This leads to the generalized continued fraction expansion lim_{n -> infinity} b(n)/a(n) = log(2) = 1/(2 - 2/(5 - 8/(8 - 18/(11 - ... - 2*(n - 1)^2/((3*n - 1) - ... ))))). (End)
%F From _Amiram Eldar_, Jun 25 2020: (Start)
%F Sum_{n>=0} 1/a(n) = sqrt(e) (A019774).
%F Sum_{n>=0} (-1)^n/a(n) = 1/sqrt(e) (A092605). (End)
%F Limit_{n->infinity} a(n)^4 / (n * A134372(n)) = Pi. - _Daniel Suteu_, Apr 09 2022
%e The following permutations and their reversals are all of the permutations of order 5 having the double-downgrade property:
%e 0 1 2 3 4
%e 0 3 2 1 4
%e 1 0 2 4 3
%e 1 4 2 0 3
%e G.f. = 1 + 2*x + 8*x^2 + 48*x^3 + 384*x^4 + 3840*x^5 + 46080*x^6 + 645120*x^7 + ...
%p A000165 := proc(n) option remember; if n <= 1 then 1 else n*A000165(n-2); fi; end;
%p 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_, Mar 26 2008
%p 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); # _Zerinvary Lajos_, Apr 03 2009
%p A000165 := proc(n) doublefactorial(2*n) ; end proc; seq(A000165(n),n=0..10) ; # _R. J. Mathar_, Oct 20 2009
%t Table[(2 n)!!, {n, 20}] (* _Vladimir Joseph Stephan Orlovsky_, Dec 13 2008 *)
%t (2 Range[0, 20])!! (* _Harvey P. Dale_, Jan 23 2015 *)
%t RecurrenceTable[{a[n] == 2 n*a[n - 1], a[0] == 1}, a, {n, 0, 19}] (* _Ray Chandler_, Jul 30 2015 *)
%o (PARI) a(n)=n!<<n \\ _Charles R Greathouse IV_, Feb 11 2011
%o (PARI) {a(n) = prod( k=1, n, 2*k)}; /* _Michael Somos_, Jan 04 2013 */
%o (Magma) [2^n*Factorial(n): n in [0..105]]; // _Vincenzo Librandi_, Apr 22 2011
%o (Magma) I:=[2,8]; [1] cat [n le 2 select I[n] else (3*n-1)*Self(n-1)-2*(n-1)^2*Self(n-2): n in [1..25] ]; // _Vincenzo Librandi_, Feb 19 2015
%o (Haskell)
%o a000165 n = product [2, 4 .. 2 * n] -- _Reinhard Zumkeller_, Mar 28 2015
%o (Python)
%o from math import factorial
%o def A000165(n): return factorial(n)<<n # _Chai Wah Wu_, Jan 24 2023
%Y Cf. A006882, A000142 (n!), A001147 ((2n-1)!!), A010050, A002454, A039683, A008544, A001813, A047053, A047055, A047058, A047657, A084947, A084948, A084949, A028326, A193229, A208057, A032184 (2^n*(n-1)!), A019774, A092605.
%Y This sequence gives the row sums in A060187, and (-1)^n*a(n) the alternating row sums in A039757. Also row sums in A028338.
%Y Column k=2 of A329070.
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_
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