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A000165 Double factorial of even numbers: (2n)!! = 2^n*n!.
(Formerly M1878 N0742)
138
1, 2, 8, 48, 384, 3840, 46080, 645120, 10321920, 185794560, 3715891200, 81749606400, 1961990553600, 51011754393600, 1428329123020800, 42849873690624000, 1371195958099968000, 46620662575398912000, 1678343852714360832000, 63777066403145711616000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

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(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, 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, 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, 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(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

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/(1-2x/(2-2x/(3-8x/(4-8x/(5-18x/(6-18x/(7-... (continued fraction). - Paul Barry, Jan 17 2011

Number of increasing 2-colored trees with choice of two colors for each edge. In general, if we replace 2 by 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

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 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

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

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

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 n-permutation. 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(m-1) is the m-th cumulant of the chi-squared probability distribution for k degrees of freedom. - Stanislav Sykora, Jun 27 2014

REFERENCES

McDonnell, Eugene, "Magic Squares and Permutations", APL Quote Quad 7.3 (Fall 1976)

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

P. Barry, General Eulerian Polynomials as Moments Using Exponential Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.9.6.

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

Eric S. Egge, Restricted symmetric permutations, Ann. Combin., 11 (2007), 405-434.

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

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

Guo-Niu Han, Enumeration of Standard Puzzles

Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]

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

Jason D. Hildebrand, Differentiating Arctan(x)

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 136

B. E. Meserve, Double Factorials, American Mathematical Monthly, 55 (1948), 425-426.

G. A. Miller, Groups formed by special matrices, Bull. Amer. Math. Soc. 24 (1918), 203-206.

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

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

Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081, 2014

R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).

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, Double Factorial

Eric Weisstein's World of Mathematics, Graph Automorphism

Index to divisibility sequences

Index entries for sequences related to factorial numbers

FORMULA

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

a(n) = 2*n * a(n-1), 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) = int(x^n*exp(-x/2)/2,x,0,infty). - Paul Barry, Jan 28 2008

G.f.: 1/(1-2x/(1-2x/(1-4x/(1-4x/(1-6x/(1-6x/(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)

G.f.: 1 + x*(Q(0) - 1)/(x+1) where Q(k)=  1 + (2*k+2)/(1 - x/(x + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Apr 11 2013

G.f.: 1/Q(0), where Q(k)= 1 + 2*k*x - 2*x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 01 2013

G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(2*k+2)/(x*(2*k+2) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 24 2013

a(n) = (2n-2)*a(n-2) + (2n-1)*a(n-1), n>1. - Ivan N. Ianakiev, Aug 06 2013

G.f.: 1/Q(0), where Q(k) = 1 - x*(4*k+2) - 4*x^2*(k+1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Sep 30 2013

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) ))); (continued fraction). - Sergei N. Gladkovskii, Oct 27 2013

EXAMPLE

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

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(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, Mar 26 2008

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

A000165 := proc(n) doublefactorial(2*n) ; end proc; seq(A000165(n), n=0..10) ; # R. J. Mathar, Oct 20 2009

MATHEMATICA

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

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, Sep 17 2008 *)

a[n_]:=(2*n)!!; (* Vladimir Joseph Stephan Orlovsky, Dec 13 2008 *)

Delete[ LinearRecurrence[{#2, #1}, {0, 1}, 40], Partition[2 Range[1, 20] - 1, 1]] (* Robert G. Wilson v, Jul 12 2014 *)

PROG

(PARI) a(n)=n!<<n \\ Charles R Greathouse IV, Feb 11 2011

(MAGMA) [2^n*Factorial(n): n in [0..105]]; // Vincenzo Librandi, Apr 22 2011

(PARI) {a(n) = prod( k=1, n, 2*k)}; /* Michael Somos, Jan 04 2013 */

CROSSREFS

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

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

This sequence gives the row sums in A060187.

Cf. A028326, A193229.

Cf. A208057. - Gary W. Adamson, Feb 22 2012

Sequence in context: A219613 A124453 A211827 * A241122 A109664 A009812

Adjacent sequences:  A000162 A000163 A000164 * A000166 A000167 A000168

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified December 20 15:01 EST 2014. Contains 252266 sequences.