

A000165


Double factorial of even numbers: (2n)!! = 2^n*n!.
(Formerly M1878 N0742)


212



1, 2, 8, 48, 384, 3840, 46080, 645120, 10321920, 185794560, 3715891200, 81749606400, 1961990553600, 51011754393600, 1428329123020800, 42849873690624000, 1371195958099968000, 46620662575398912000, 1678343852714360832000, 63777066403145711616000
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OFFSET

0,2


COMMENTS

a(n) is also the size of the automorphism group of the graph (edge graph) of the ndimensional 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, Feb 02 2002
Appears to be the BinomialMean transform of A001907. See A075271.  John W. Layman, Sep 28 2002
Number of n X n monomial matrices with entries 0, +1.
a(n) = A001044(n)/A000142(n)*A000079(n) = Product_{i=0..n1} (2*i+2) = 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 doubledowngrades. The number of permutations of order 2n having this property are equinumerous with those of order 2n+1. a(n) = number of doubledowngrading permutations of order 2n and 2n+1.  Eugene McDonnell (eemcd(AT)mac.com), Oct 27 2003
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
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, Jan 14 2006
a(n) is the Pfaffian of the skewsymmetric 2n X 2n matrix whose (i,j) entry is j for i<j.  David Callan, 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 012 trees.  David Callan, 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, 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, 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(n1). The mapping p > (i,q) is a bijection from A(n) to the Cartesian product [1,2n] X A(n1).  David Callan, Nov 29 2007
Row sums of A028338.  Paul Barry, Feb 07 2009
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
From Gary W. Adamson, Apr 21 2009: (Start)
Equals (1)^n * (1, 1, 2, 8, 48, ...) dot (1, 3, 5, 7, 9, ...).
Example: a(4) = 384 = (1, 1, 2, 8, 48) dot (1, 3, 5, 7, 9) = (1, 3, 10, 56, 432). (End)
exp(x/2)=sum(n>=0,x^n/a(n)).  Jaume Oliver Lafont, Sep 07 2009
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
E.g.f. of 0,1,2,8,... is x/(12x/(22x/(38x/(48x/(518x/(618x/(7... (continued fraction).  Paul Barry, Jan 17 2011
Number of increasing 2colored trees with choice of two colors for each edge. In general, if we replace 2 with k we get the number of increasing kcolored trees. For example, for k=3 we get the triple factorial numbers.  Wenjin Woan, May 31 2011
a(n) = row sums of triangle A193229.  Gary W. Adamson, Jul 18 2011
Also the number of permutations of 2n (or of 2n+1) that are equal to their reversecomplements. (See the Egge reference.) Note that the doubledowngrade described in the preceding comment (McDonnell) is equivalent to the reversecomplement.  Justin M. Troyka, Aug 11 2011
The e.g.f. can be used to form a generator, [1/(12x)] d/dx, for A000108, so a(n) can be applied to A145271 to generate the Catalan numbers.  Tom Copeland, Oct 01 2011
The e.g.f. of 1/a(n) is BesselI(0,sqrt(2*x)). See AbramowitzStegun (reference and link under A008277), p. 375, 9.6.10.  Wolfdieter Lang, Jan 09 2012
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
Row sums of triangle A208057.  Gary W. Adamson, Feb 22 2012
a(n) is the number of ways to designate a subset of elements in each npermutation. a(n) = A000142(n) + A001563(n) + A001804(n) + A001805(n) + A001806(n) + A001807(n) + A035038(n) * n!.  Geoffrey Critzer, Nov 08 2012
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
For m>0, k*a(m1) is the mth cumulant of the chisquared probability distribution for k degrees of freedom.  Stanislav Sykora, Jun 27 2014
a(n) with 0 prepended is the binomial transform of A120765.  Vladimir Reshetnikov, Oct 28 2015
Exponential selfconvolution of A001147.  Vladimir Reshetnikov, Oct 08 2016
Also the order of the automorphism group of the nladder rung graph.  Eric W. Weisstein, Jul 22 2017
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


REFERENCES

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

T. D. Noe, Table of n, a(n) for n = 0..100
Paul Barry, General Eulerian Polynomials as Moments Using Exponential Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.9.6.
Paul Barry, Generalized Eulerian Triangles and Some Special Production Matrices, arXiv:1803.10297 [math.CO], 2018.
Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, Combinatorial Identities Associated with a Multidimensional Polynomial Sequence, J. Int. Seq., Vol. 21 (2018), Article 18.7.4.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
CombOS  Combinatorial Object Server, Generate colored permutations
R. Coquereaux and J.B. Zuber, Maps, immersions and permutations, arXiv preprint arXiv:1507.03163 [math.CO], 2015.
Eric S. Egge, Restricted symmetric permutations, Ann. Combin., 11 (2007), 405434.
Peter C. Fishburn, Signed Orders, Choice Probabilities and Linear Polytopes, Journal of Mathematical Psychology, Volume 45, Issue 1, (2001), pp. 5380.
Joël Gay and Vincent Pilaud, The weak order on Weyl posets, arXiv:1804.06572 [math.CO], 2018.
G. Gordon, The answer is 2^n*n! What is the question?, Amer. Math. Monthly, 106 (1999), 636645.
GuoNiu Han, Enumeration of Standard Puzzles
GuoNiu Han, Enumeration of Standard Puzzles [Cached copy]
Hamed Hatami and Pooya Hatami, Perfect dominating sets in the Cartesian products of prime cycles, arXiv:math/0701018 [math.CO], 20062009.
Jason D. Hildebrand, Differentiating Arctan(x)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 136
L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180181. [Annotated scan of pages 180 and 181 only]
E. Lucas, Théorie des nombres (annotated scans of a few selected pages)
Eugene McDonnell, Magic Squares and Permutations, APL Quote Quad 7.3 (Fall 1976).
B. E. Meserve, Double Factorials, American Mathematical Monthly, 55 (1948), 425426.
G. A. Miller, Groups formed by special matrices, Bull. Amer. Math. Soc. 24 (1918), 203206.
R. Ondrejka, Tables of double factorials, Math. Comp., 24 (1970), 231.
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some WellKnown Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.
Luis Manuel Rivera, Integer sequences and kcommuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
R. W. Robinson, Counting arrangements of bishops, pp. 198214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
M. Z. Spivey and L. L. Steil, The kBinomial Transforms and the Hankel Transform, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.
Eric Weisstein's World of Mathematics, Double Factorial
Eric Weisstein's World of Mathematics, Graph Automorphism
Eric Weisstein's World of Mathematics, Ladder Rung Graph
Index to divisibility sequences
Index entries for sequences related to factorial numbers


FORMULA

E.g.f.: 1/(12*x).
Dfinite with recurrence a(n) = 2*n * a(n1), n>0, a(0)=1.  Paul Barry, 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) = Integral_{x>=0} x^n*exp(x/2)/2 dx.  Paul Barry, Jan 28 2008
G.f.: 1/(12x/(12x/(14x/(14x/(16x/(16x/(1.... (continued fraction).  Paul Barry, Feb 07 2009
a(n) = A006882(2*n).  R. J. Mathar, Oct 20 2009
From Gary W. Adamson, Jul 18 2011: (Start)
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):
2, 2, 0, 0, 0, 0, ...
2, 4, 2, 0, 0, 0, ...
2, 6, 6, 2, 0, 0, ...
2, 8, 12, 8, 2, 0, ...
2, 10, 20, 20, 10, 2, ...
... (End)
From Sergei N. Gladkovskii, Apr 11 2013, May 01 2013, May 24 2013, Sep 30 2013, Oct 27 2013: (Start)
Continued fractions:
G.f.: 1 + x*(Q(0)  1)/(x+1) where Q(k) = 1 + (2*k+2)/(1x/(x+1/Q(k+1))).
G.f.: 1/Q(0) where Q(k) = 1 + 2*k*x  2*x*(k+1)/Q(k+1).
G.f.: G(0)/2 where G(k) = 1 + 1/(1  x*(2*k+2)/(x*(2*k+2) + 1/G(k+1))).
G.f.: 1/Q(0) where Q(k) = 1  x*(4*k+2)  4*x^2*(k+1)^2/Q(k+1).
G.f.: R(0) where R(k) = 1  x*(2*k+2)/(x*(2*k+2)1/(1x*(2*k+2)/(x*(2*k+2) 1/R(k+1)))). (End)
a(n) = (2n2)*a(n2) + (2n1)*a(n1), n>1.  Ivan N. Ianakiev, Aug 06 2013
From Peter Bala, Feb 18 2015: (Start)
Recurrence equation: a(n) = (3*n  1)*a(n1)  2*(n  1)^2*a(n2) with a(1) = 2 and a(2) = 8.
The sequence b(n) = A068102(n) also satisfies this secondorder recurrence. This leads to the generalized continued fraction expansion lim_{n > inf} b(n)/a(n) = log(2) = 1/(2  2/(5  8/(8  18/(11  ...  2*(n  1)^2/((3*n  1)  ... ))))). (End)
From Amiram Eldar, Jun 25 2020: (Start)
Sum_{n>=0) 1/a(n) = sqrt(e) (A019774).
Sum_{n>=0) (1)^n/a(n) = 1/sqrt(e) (A092605). (End)


EXAMPLE

The following permutations and their reversals are all of the permutations of order 5 having the doubledowngrade property:
0 1 2 3 4
0 3 2 1 4
1 0 2 4 3
1 4 2 0 3
G.f. = 1 + 2*x + 8*x^2 + 48*x^3 + 384*x^4 + 3840*x^5 + 46080*x^6 + 645120*x^7 + ...


MAPLE

A000165 := proc(n) option remember; if n <= 1 then 1 else n*A000165(n2); 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, Mar 26 2008
G(x):=(12*x)^(1): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n1], x) od: x:=0: seq(f[n], n=0..17); # Zerinvary Lajos, Apr 03 2009
A000165 := proc(n) doublefactorial(2*n) ; end proc; seq(A000165(n), n=0..10) ; # R. J. Mathar, Oct 20 2009


MATHEMATICA

Table[(2 n)!!, {n, 20}] (* Vladimir Joseph Stephan Orlovsky, Dec 13 2008 *)
(2 Range[0, 20])!! (* Harvey P. Dale, Jan 23 2015 *)
RecurrenceTable[{a[n] == 2 n*a[n  1], a[0] == 1}, a, {n, 0, 19}] (* Ray Chandler, Jul 30 2015 *)


PROG

(PARI) a(n)=n!<<n \\ Charles R Greathouse IV, Feb 11 2011
(PARI) {a(n) = prod( k=1, n, 2*k)}; /* Michael Somos, Jan 04 2013 */
(MAGMA) [2^n*Factorial(n): n in [0..105]]; // Vincenzo Librandi, Apr 22 2011
(MAGMA) I:=[2, 8]; [1] cat [n le 2 select I[n] else (3*n1)*Self(n1)2*(n1)^2*Self(n2): n in [1..25] ]; // Vincenzo Librandi, Feb 19 2015
(Haskell)
a000165 n = product [2, 4 .. 2 * n]  Reinhard Zumkeller, Mar 28 2015


CROSSREFS

Cf. A006882, A000142 (n!), A001147 ((2n1)!!), A010050, A002454, A039683, A008544, A001813, A047053, A047055, A047058, A047657, A084947, A084948, A084949, A028326, A193229, A208057, A032184 (2^n*(n1)!), A019774, A092605.
This sequence gives the row sums in A060187, and (1)^n*a(n) the alternating row sums in A039757. Also row sums in A028338.
Column k=2 of A329070.
Sequence in context: A124453 A211827 A322339 * A241122 A109664 A009812
Adjacent sequences: A000162 A000163 A000164 * A000166 A000167 A000168


KEYWORD

nonn,easy,nice


AUTHOR

N. J. A. Sloane


STATUS

approved



