OFFSET
0,5
COMMENTS
a(n) is the number of ways of separating n terms into pairs. - Stephen Crowley, Apr 07 2007
a(n) is the n-th moment of the standard normal distribution. - Hal M. Switkay, Nov 06 2019
a(n) is the number of fixed-point free involutions in the symmetric group of degree n. - Nick Krempel, Feb 26 2020
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 = 0..799
Mihai Nica, Terence Tao's central limit theorem, Double Factorials (!!) and the Moment Method, YouTube video (2023).
Sebastian Volz, Design and Implementation of Efficient Algorithms for Operations on Partitions of Sets, Bachelor Thesis, Saarland Univ. (Germany, 2023). See p. 45.
Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017-2018, Table 2 on p. 15.
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
E.g.f.: exp(x^2/2). - Geoffrey Critzer, Mar 15 2009
a(2n) = A001147(n). - 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.: 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/G(0) where G(k) = 1 - x^2*(k+1)/G(k+1).
G.f.: 1 + x^2/(1+x) + Q(0)*x^3/(1+x), where Q(k) = 1 + (2*k+3)*x/(1 - x/(x + 1/Q(k+1))).
G.f.: 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/(1+x) + 1, where G(k) = 1 + x*(2*k+1)/(1 - x/(x + 1/G(k+1))). (End)
For n even, a(n) = A001147(n/2) = A124794(3^(n/2)). a(n) is also the coefficient of x1*...*xn in Product_{1 <= i < j <= n} (1 + xi*xj). - Gus Wiseman, Dec 23 2018
a(n) = 2^(n/2)*Pochhammer(1/2, n/2)*(n+1 mod 2). - Peter Luschny, Jan 11 2023
EXAMPLE
From Gus Wiseman, Dec 23 2018: (Start)
The a(6) = 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=0..36); # Zerinvary Lajos, Sep 24 2007
a := n -> ifelse(irem(n, 2) = 1, 0, 2^(n/2) * pochhammer(1/2, n/2)):
seq(a(n), n = 0..36); # Peter Luschny, Jan 11 2023
MATHEMATICA
RecurrenceTable[{a[0] == 1, a[1] == 0, a[n] == (n - 1) a[n - 2]}, a[n], {n, 0, 31}] (* Ray Chandler, Jul 30 2015 *)
PROG
(Magma) a:=[1, 0]; [n le 2 select a[n] else (n-2)*Self(n-2): n in [1..30]]; // Marius A. Burtea, Nov 07 2019
CROSSREFS
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
Leading term 1 dropped, offset changed, and entry edited correspondingly by Andrey Zabolotskiy, Nov 07 2019
STATUS
approved