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A006882 Double factorials n!!: a(n)=n*a(n-2), a(0)=a(1)=1.
(Formerly M0876)
154
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 (list; graph; refs; listen; history; text; internal format)
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

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

T. D. Noe, Table of n, a(n) for n = 0..100

P. Luschny, Multifactorials

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

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

Eric Weisstein, Double Factorial , (MathWorld)

Eric Weisstein, Multifactorial, (MathWorld)

Index entries for sequences related to factorial numbers

Index entries for "core" sequences

FORMULA

a(n) = prod(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

EXAMPLE

1 + x + 2*x^2 + 3*x^3 + 8*x^4 + 15*x^5 + 48*x^6 + 105*x^7 + 384*x^8 + 945*x^9 + ...

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

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, local(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

CROSSREFS

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

Cf. A052319.

A diagonal of A202212.

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

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Last modified October 20 17:08 EDT 2014. Contains 248350 sequences.