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 A072187 Number of up-down involutions of length n. 2
 1, 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, 923739847132, 3550638576721, 13885783706324 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS This resulted from a question from Richard Ehrenborg and Margie Readdy. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..500 (terms n = 1..50 from Vladeta Jovovic) D. Zeilberger, I Am Sorry, Richard Ehrenborg and Margie Readdy, About Your Two Conjectures, But One Is FAMOUS, While The Other Is FALSE; Local copy [PDF file only, no active links] FORMULA G.f.: Sum_{n>=0} a(2n+1)x^(2n+1) = Sum_{i,j >= 0) 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 an Euler number (A000111). There is a similar but more complicated generating function for a(2n). - Richard Stanley, Jan 02 2006 EXAMPLE a(3)=1 since among the four involutions of length 3 (123, 213, 321, 132), only one is up-down (132). CROSSREFS Cf. A000085, A000111. Sequence in context: A000992 A036648 A047750 * A072374 A122852 A192573 Adjacent sequences: A072184 A072185 A072186 * A072188 A072189 A072190 KEYWORD nonn AUTHOR Doron Zeilberger, Jul 01 2002; more terms, Dec 09 2003 EXTENSIONS More terms from Vladeta Jovovic, May 16 2007 a(0)=1 prepended by Alois P. Heinz, Aug 07 2018 STATUS approved

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