0,3

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

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).

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

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Index entries for sequences related to factorial numbers

Index entries for "core" sequences

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 (ppl(AT)spl.at), 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 (ppl(AT)spl.at), Jul 24 2007

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]

Array[ #!!&, 40, 0 ]

(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) [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 */

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

Cf. A052319.

Sequence in context: A148011 A148012 A161178 this_sequence A080498 A148013 A133983

Adjacent sequences: A006879 A006880 A006881 this_sequence A006883 A006884 A006885

nonn,easy,core,nice

mrob(AT)mrob.com (Robert P Munafo)