A072187 Number of up-down involutions of length n.

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

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