A006882 Double factorials n!!: a(n) = n*a(n-2) for n > 1, a(0) = a(1) = 1. (Formerly M0876)

1, 1, 2, 3, 8, 15, 48, 105, 384, 945, 3840, 10395, 46080, 135135, 645120, 2027025, 10321920, 34459425, 185794560, 654729075, 3715891200, 13749310575, 81749606400, 316234143225, 1961990553600, 7905853580625, 51011754393600, 213458046676875, 1428329123020800

Offset

0,3

Comments

Product of pairs of successive terms gives factorials in increasing order. - Amarnath Murthy, Oct 17 2002

a(n) = number of down-up permutations on [n+1] for which the entries in the even positions are increasing. For example, a(3)=3 counts 2143, 3142, 4132. Also, a(n) = number of down-up permutations on [n+2] for which the entries in the odd positions are decreasing. For example, a(3)=3 counts 51423, 52413, 53412. - David Callan, Nov 29 2007

The double factorial of a positive integer n is the product of the positive integers <= n that have the same parity as n. - Peter Luschny, Jun 23 2011

For n even, a(n) is the number of ways to place n points on an n X n grid with pairwise distinct abscissa, pairwise distinct ordinate, and 180-degree rotational symmetry. For n odd, the number of ways is a(n-1) because the center point can be considered "fixed". For 90-degree rotational symmetry cf. A001813, for mirror symmetry see A000085, A135401, and A297708. - Manfred Scheucher, Dec 29 2017

Could be extended to include a(-1) = 1. But a(-2) is not defined, otherwise we would have 1 = a(0) = 0*a(-2). - Jianing Song, Oct 23 2019

References

Putnam Contest, 4 Dec. 2004, Problem A3.

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

Links

Indranil Ghosh, Table of n, a(n) for n = 0..806 (terms 0..100 from T. D. Noe)

Christian Aebi and Grant Cairns, Generalizations of Wilson's Theorem for Double-, Hyper-, Sub-and Superfactorials, The American Mathematical Monthly 122.5 (2015): 433-443.

CombOS - Combinatorial Object Server, Generate colored permutations

Joseph E. Cooper III, A recurrence for an expression involving double factorials, arXiv:1510.00399 [math.CO], 2015.

Gary T. Leavens and Mike Vermeulen, 3x+1 search programs, Computers and Mathematics with Applications, 24 (1992), 79-99. (Annotated scanned copy)

Peter Luschny, Multifactorials

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

Rudolph Ondrejka, Tables of double factorials, Math. Comp., Vol. 24, No. 109 (1970), p. 231.

Eric Weisstein's World of Mathematics, Double Factorial.

Eric Weisstein's World of Mathematics, Multifactorial.

Index entries for sequences related to factorial numbers

Index entries for "core" sequences

Formula

a(n) = Product_{i=0..floor((n-1)/2)} (n - 2*i).

E.g.f.: 1+exp(x^2/2)*x*(1+sqrt(Pi/2)*erf(x/sqrt(2))). - Wouter Meeussen, Mar 08 2001

Satisfies a(n+3)*a(n) - a(n+1)*a(n+2) = (n+1)!. [Putnam Contest]

n!! = 2^[(n + 1)/2]/sqrt(Pi)*Gamma(n/2 + 1)*{[sqrt(Pi)/2^(1/2) + 1]/2 + (-1)^n*[sqrt(Pi)/2^(1/2)-1]/2}. - Paolo P. Lava, Jul 24 2007

a(n) = 2^((1+2*n-cos(n*Pi))/4)*Pi^((cos(n*Pi)-1)/4)*Gamma(1+(1/2)*n). - Paolo P. Lava, Jul 24 2007

a(n) = n!/a(n-1). - Vaclav Kotesovec, Sep 17 2012

a(n) * a(n+3) = a(n+1) * (a(n+2) + a(n)). a(n) * a(n+1) = (n+1)!. - Michael Somos, Dec 29 2012

a(n) ~ c * n^((n+1)/2) / exp(n/2), where c = sqrt(Pi) if n is even, and c = sqrt(2) if n is odd. - Vaclav Kotesovec, Nov 08 2014

a(2*n) = 2^n*a(n)*a(n-1). a(2^n) = 2^(2^n - 1) * 1!! * 3!! * 7!! * ... * (2^(n-1) - 1)!!. - Peter Bala, Nov 01 2016

a(n) = 2^h*(2/Pi)^(sin(Pi*h)^2/2)*Gamma(h+1) where h = n/2. This analytical extension supports the view that a(-1) = 1 is a meaningful numerical extension. With this definition (-1/2)!! = Gamma(3/4)/Pi^(1/4). - Peter Luschny, Oct 24 2019

a(n) ~ (n+1/6)*sqrt((2/e)*(n/e)^(n-1)*(Pi/2)^(cos(n*Pi/2)^2)). - Peter Luschny, Oct 25 2019

Sum_{n>=0} 1/a(n) = A143280. - Amiram Eldar, Nov 10 2020

Sum_{n>=0} 1/(a(n)*a(n+1)) = e - 1. - Andrés Ventas, Apr 12 2021

Example

G.f. = 1 + x + 2*x^2 + 3*x^3 + 8*x^4 + 15*x^5 + 48*x^6 + 105*x^7 + 384*x^8 + ...

Maple

A006882 := proc(n) option remember; if n <= 1 then 1 else n*A006882(n-2); fi; end;
A006882 := proc(n) doublefactorial(n) ; end proc; seq(A006882(n), n=0..10) ; # R. J. Mathar, Oct 20 2009
A006882 := n -> mul(k, k = select(k -> k mod 2 = n mod 2, [$1 .. n])):  seq(A006882(n), n = 0 .. 10); # Peter Luschny, Jun 23 2011
A006882 := proc(n) if n=0 then 1 else mul(n-2*k, k=0..floor(n/2)-1); fi; end; # N. J. A. Sloane, May 27 2016

Mathematica

Array[ #!!&, 40, 0 ]
multiFactorial[n_, k_] := If[n < 1, 1, If[n < k + 1, n, n*multiFactorial[n - k, k]]]; Array[ multiFactorial[#, 2] &, 27, 0] (* Robert G. Wilson v, Apr 23 2011 *)

Prog

(PARI) {a(n) = prod(i=0, (n-1)\2, n - 2*i )} \\ Improved by M. F. Hasler, Nov 30 2013
(PARI) {a(n) = if( n<2, n>=0, n * a(n-2))}; /* Michael Somos, Apr 06 2003 */
(PARI) {a(n) = if( n<0, 0, my(E); E = exp(x^2 / 2 + x * O(x^n)); n! * polcoeff( 1 + E * x * (1 + intformal(1 / E)), n))}; /* Michael Somos, Apr 06 2003 */
(Magma) DoubleFactorial:=func< n | &*[n..2 by -2] >; [ DoubleFactorial(n): n in [0..28] ]; // Klaus Brockhaus, Jan 23 2011
(Haskell)
a006882 n = a006882_list !! n
a006882_list = 1 : 1 : zipWith (*) [2..] a006882_list
-- Reinhard Zumkeller, Oct 23 2014
(Python)
from sympy import factorial2
def A006882(n): return factorial2(n) # Chai Wah Wu, Apr 03 2021

Crossrefs

Bisections are A000165 and A001147. These two entries have more information.

Cf. A052319, A143280.

A diagonal of A202212.

Cf. A000085, A001813, A135401, A297708. - Manfred Scheucher, Jan 07 2018

Sequence in context: A148011 A148012 A161178 * A080498 A148013 A133983

Adjacent sequences: A006879 A006880 A006881 * A006883 A006884 A006885

Keyword

nonn,easy,core,nice

Author

Robert Munafo

Status

approved