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A006882
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Double factorials n!!: a(n)=n*a(n-2), a(0)=a(1)=1.
(Formerly M0876)
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120
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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; internal format)
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OFFSET
| 0,3
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COMMENTS
| Product of pairs of successive terms gives factorials in increasing order. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), 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 (callan(AT)stat.wisc.edu), 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
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REFERENCES
| B. E. Meserve, Double Factorials, American Mathematical Monthly, 55 (1948), 425-426.
R. Ondrejka, Tables of double factorials, Math. Comp., 24 (1970), 231.
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).
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LINKS
| T. D. Noe, Table of n, a(n) for n = 0..100
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
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FORMULA
| a(n) = prod(i=0..floor((n-1)/2), n-2*i )
E.g.f.: 1+e^(x^2/2) x (1+Sqrt[Pi/2] Erf[x/Sqrt[2]]) - wouter.meeussen(AT)pandora.be Thu Mar 08 07:17:05 2001
Satisfies a(n+3)*a(n) - a(n+1)*a(n+2) = n! [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 (paoloplava(AT)gmail.com), 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 (paoloplava(AT)gmail.com), Jul 24 2007
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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) ; [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 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
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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 *)
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PROG
| (PARI) a(n) = prod(i=0, floor((n-1)/2), n-2*i );
(PARI) a(n)=if(n<2, 1, n*a(n-2))
(PARI) a(n)=local(E); E=exp(x^2/2+x*O(x^n)); n!*polcoeff(1+E*x*(1+intformal(1/E)), n)
(MAGMA) DoubleFactorial:=func< n | &*[n..2 by -2] >; [ DoubleFactorial(n): n in [0..28] ]; - Klaus Brockhaus, Jan 23 2011
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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
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KEYWORD
| nonn,easy,core,nice
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AUTHOR
| mrob(AT)mrob.com (Robert P Munafo)
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