This site is supported by donations to The OEIS Foundation.

Thanks to everyone who made a donation during our annual appeal!
To see the list of donors, or make a donation, see the OEIS Foundation home page.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A123023 a(n) = (n-2)*a(n-2), a(0)=1, a(1)=1. 13
 1, 1, 0, 1, 0, 3, 0, 15, 0, 105, 0, 945, 0, 10395, 0, 135135, 0, 2027025, 0, 34459425, 0, 654729075, 0, 13749310575, 0, 316234143225, 0, 7905853580625, 0, 213458046676875, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS Old name was: a(n) = b(n)*n!, where b(n+2) = n*b(n)/((n+2)*(n+1)), b(0) = 0, b(1) = 1. a(n > 1) is the number of ways of separating n - 2 terms into pairs. - Stephen Crowley, Apr 07 2007 REFERENCES Richard Bronson, Schaum's Outline of Modern Introductory Differential Equations, MacGraw-Hill, New York,1973, page 107, solved problem 19.18 Norbert Wiener, Nonlinear Problems in Random Theory, 1958, Equation 1.31 LINKS G. C. Greubel, Table of n, a(n) for n = 1..800 Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017, Table 2 on p. 13. FORMULA a(n) = (1/2)*GAMMA((1/2)*n + 1/2)*2^((1/2)*n)*(1 + (-1)^n)/sqrt(Pi). - Stephen Crowley, Apr 07 2007 With offset -1, E.g.f.: exp(x^2/2). - Geoffrey Critzer, Mar 15 2009 a(2n) = A001147(n-1). - R. J. Mathar, Oct 11 2011 From Sergei N. Gladkovskii, Nov 18 2012, Dec 05 2012, May 16 2013, May 24 2013, Jun 07 2013: (Start) Continued fractions: E.g.f.: 1 + x*E(0) where E(k) = 1 + x^2*(4*k+1)/((4*k+2)*(4*k+3) - x^2*(4*k+2)*(4*k+3)^2/(x^2*(4*k+3) + (4*k+4)*(4*k+5)/E(k+1))). G.f.: 1 + x/G(0) where G(k) =  1 - x^2*(k+1)/G(k+1). G.f.: 1 + x + x^3/(1+x) + Q(0)*x^4/(1+x), where Q(k) = 1 + (2*k+3)*x/(1 - x/(x + 1/Q(k+1))). G.f.: 1 + x*G(0)/2, where G(k) = 1 + 1/(1-x/(x+1/x/(2*k+1)/G(k+1))). G.f.: (G(0) - 1)*x^2/(1+x) + 1 + x, where G(k) = 1 + x*(2*k+1)/(1 - x/(x + 1/G(k+1))). (End) For n even, a(n + 2) = A001147(n/2) = A124794(3^(n/2)). a(n + 2) is also the coefficient of x1*...*xk in Product_{1 <= i < j <= n} (1 + xi*xj). - Gus Wiseman, Dec 23 2018 EXAMPLE From Gus Wiseman, Dec 23 2018: (Start) The a(8) = 15 ways of partitioning {1,2,3,4,5,6} into disjoint pairs:   {{12}{34}{56}}   {{12}{35}{46}}   {{12}{36}{45}}   {{13}{24}{56}}   {{13}{25}{46}}   {{13}{26}{45}}   {{14}{23}{56}}   {{14}{25}{36}}   {{14}{26}{35}}   {{15}{23}{46}}   {{15}{24}{36}}   {{15}{26}{34}}   {{16}{23}{45}}   {{16}{24}{35}}   {{16}{25}{34}} (End) MAPLE with(combstruct):ZL2:=[S, {S=Set(Cycle(Z, card=2))}, labeled]:seq(count(ZL2, size=n), n=1..28); # Zerinvary Lajos, Sep 24 2007 MATHEMATICA b[n_] := b[n] = (n - 2) * b[n - 2]/(n * (n - 1)); b[0] = 1; b[1] = 1; Table[b[n] * n!, {n, 0, 30}] RecurrenceTable[{a[0] == 1, a[1] == 1, a[n] == (n - 2) a[n - 2]}, a[n], {n, 0, 31}] (* Ray Chandler, Jul 30 2015 *) CROSSREFS Cf. A000085, A000110, A000124, A000142, A000587, A001147, A001464, A006129, A079267, A124794, A186021, A295193. Sequence in context: A167339 A277936 A138540 * A130637 A054882 A303232 Adjacent sequences:  A123020 A123021 A123022 * A123024 A123025 A123026 KEYWORD nonn,easy AUTHOR Roger L. Bagula, Sep 24 2006 EXTENSIONS Edited by N. J. A. Sloane, Jan 06 2008 Better name by Sergei N. Gladkovskii, May 24 2013 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified January 16 21:37 EST 2019. Contains 319206 sequences. (Running on oeis4.)