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A072187
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Number of up-down involutions of length n.
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1
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1, 1, 1, 2, 3, 6, 11, 24, 51, 120, 283, 716, 1833, 4948, 13561, 38788, 112745, 339676, 1039929, 3283876, 10532747, 34717276, 116158851, 398257012, 1385117947, 4925094508, 17752742867, 65297807204, 243319812785
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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COMMENTS
| This resulted from a question from Richard Ehrenborg and Margie Readdy.
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LINKS
| Vladeta Jovovic, Table of n, a(n) for n = 1..50
D. Zeilberger, I Am Sorry, Richard Ehrenborg and Margie Readdy, About Your Two Conjectures, But One Is FAMOUS, While The Other Is FALSE
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FORMULA
| G.f.: Sum_{n=0..infinity} a(2n+1)x^(2n+1) = Sum_{i, j=0..infinity) arctan(x)^(2i+1) (log((1+x^2)/(1-x^2)))^j E(2i+2j+1)/(2i+1)!j!4^j, where E(2i+2j+1) is a Euler number (A000111). There is a similar but more complicated generating function for a(2n). - R. P. Stanley (rstan(AT)math.mit.edu), Jan 02 2006
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EXAMPLE
| a(3)=1 since among the four involutions of length 3 (123, 213, 321, 132), only one is up-down (132).
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CROSSREFS
| Cf. A000111.
Sequence in context: A000992 A036648 A047750 * A122852 A072374 A192573
Adjacent sequences: A072184 A072185 A072186 * A072188 A072189 A072190
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KEYWORD
| nonn
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AUTHOR
| Doron Zeilberger (zeilberg(AT)math.rutgers.edu), Jul 01 2002; more terms, Dec 09 2003
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EXTENSIONS
| More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), May 16 2007
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