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

Author

Doron Zeilberger, Jul 01 2002; more terms, Dec 09 2003

Keywords

• nonn

Comments

This resulted from a question from Richard Ehrenborg and Margie Readdy.

Examples

			a(3)=1 since among the four involutions of length 3 (123, 213, 321, 132), only one is up-down (132).
		

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]

Crossrefs

Cf. A000085, A000111.

Formula

G.f.: Sum_{n>=0} a(2*n+1)*x^(2*n+1) = Sum_{i,j >= 0} arctan(x)^(2*i+1)*(log((1+x^2)/(1-x^2)))^j*E(2*i+2*j+1)/((2*i+1)!*j!*4^j), where E(2*i+2*j+1) is an Euler number (A000111). There is a similar but more complicated generating function for a(2*n). - Richard Stanley, Jan 02 2006

Extensions

More terms from Vladeta Jovovic, May 16 2007

a(0)=1 prepended by Alois P. Heinz, Aug 07 2018