|
| |
|
|
A006882
|
|
Double factorials n!!: a(n)=n*a(n-2), a(0)=a(1)=1.
(Formerly M0876)
|
|
157
|
|
|
|
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
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
|
|
|
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
(Haskell)
a006882 n = a006882_list !! n
a006882_list = 1 : 1 : zipWith (*) [2..] a006882_list
-- Reinhard Zumkeller, Oct 23 2014
|
|
|
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
|
| |
|
|
