Search: id:A006882
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%I A006882 M0876
%S A006882 1,1,2,3,8,15,48,105,384,945,3840,10395,46080,135135,645120,2027025,
%T A006882 10321920,34459425,185794560,654729075,3715891200,13749310575,
%U A006882 81749606400,316234143225,1961990553600,7905853580625,51011754393600
%N A006882 Double factorials n!!: a(n)=n*a(n-2).
%C A006882 Product of pairs of successive terms gives factorials in increasing order. 
               - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Oct 17 2002
%C A006882 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
%D A006882 B. E. Meserve, Double Factorials, American Mathematical Monthly, 55 (1948), 
               425-426.
%D A006882 R. Ondrejka, Tables of double factorials, Math. Comp., 24 (1970), 231.
%D A006882 Putnam Contest, 4 Dec. 2004, Problem A3.
%D A006882 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A006882 T. D. Noe, <a href="b006882.txt">Table of n, a(n) for n = 0..100</a>
%H A006882 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               DoubleFactorial.html">Link to a section of The World of Mathematics.</
               a>
%H A006882 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Multifactorial.html">Link to a section of The World of Mathematics.</
               a>
%H A006882 <a href="Sindx_Fa.html#factorial">Index entries for sequences related 
               to factorial numbers</a>
%H A006882 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A006882 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
%F A006882 Satisfies a(n+3)*a(n) - a(n+1)*a(n+2) = n! [Putnam Contest]
%F A006882 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 (ppl(AT)spl.at), 
               Jul 24 2007
%F A006882 a(n)=2^{[1+2*n-cos(n*Pi)]/4}*Pi^{[cos(n*Pi)-1]/4}*Gamma(1+1/2*n) - Paolo 
               P. Lava (ppl(AT)spl.at), Jul 24 2007
%p A006882 A006882 := proc(n) option remember; if n <= 1 then 1 else n*A006882(n-2); 
               fi; end;
%p A006882 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]
%t A006882 Array[ #!!&, 40, 0 ]
%o A006882 (PARI) a(n)=if(n<2,1,n*a(n-2))
%o A006882 (PARI) a(n)=local(E); E=exp(x^2/2+x*O(x^n)); n!*polcoeff(1+E*x*(1+intformal(1/
               E)),n)
%o A006882 (MAGMA) [1] cat &cat[ [ &*[ 2*k+1: k in [0..n] ], &*[ 2*(k+1): k in [0..n] 
               ] ]: n in [0..12] ]; /* Klaus Brockhaus, Apr 14 2007 */
%Y A006882 Bisections are A000165 and A001147. These two entries have much more 
               information.
%Y A006882 Cf. A052319.
%Y A006882 Sequence in context: A148011 A148012 A161178 this_sequence A080498 A148013 
               A133983
%Y A006882 Adjacent sequences: A006879 A006880 A006881 this_sequence A006883 A006884 
               A006885
%K A006882 nonn,easy,core,nice
%O A006882 0,3
%A A006882 mrob(AT)mrob.com (Robert P Munafo)

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