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A123023
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a(n) = (n-1)*a(n-2), a(0)=1, a(1)=0.
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22
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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, 6190283353629375, 0, 191898783962510625, 0, 6332659870762850625, 0, 221643095476699771875
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OFFSET
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0,5
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COMMENTS
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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
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REFERENCES
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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
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LINKS
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FORMULA
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a(n) = (1/2)*Gamma((1/2)*n + 1/2)*2^((1/2)*n)*(1 + (-1)^n)/sqrt(Pi). - Stephen Crowley, Apr 07 2007
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
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EXAMPLE
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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)
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MAPLE
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with(combstruct): ZL2 := [S, {S=Set(Cycle(Z, card=2))}, labeled]:
a := n -> ifelse(irem(n, 2) = 1, 0, 2^(n/2) * pochhammer(1/2, n/2)):
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MATHEMATICA
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RecurrenceTable[{a[0] == 1, a[1] == 0, a[n] == (n - 1) a[n - 2]}, a[n], {n, 0, 31}] (* Ray Chandler, Jul 30 2015 *)
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PROG
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(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
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CROSSREFS
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Cf. A000085, A000110, A000124, A000142, A000587, A001147, A001464, A006129, A079267, A124794, A186021, A295193.
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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Leading term 1 dropped, offset changed, and entry edited correspondingly by Andrey Zabolotskiy, Nov 07 2019
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STATUS
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approved
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