This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A072187 #43 Apr 20 2025 20:12:56
%S A072187 1,1,1,1,2,3,6,11,24,51,120,283,716,1833,4948,13561,38788,112745,
%T A072187 339676,1039929,3283876,10532747,34717276,116158851,398257012,
%U A072187 1385117947,4925094508,17752742867,65297807204,243319812785,923739847132,3550638576721,13885783706324
%N A072187 Number of up-down involutions of length n.
%C A072187 This resulted from a question from _Richard Ehrenborg_ and Margie Readdy.
%H A072187 Alois P. Heinz, <a href="/A072187/b072187.txt">Table of n, a(n) for n = 0..500</a> (terms n = 1..50 from Vladeta Jovovic)
%H A072187 D. Zeilberger, <a href="https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/ehrenborg.html">I Am Sorry, Richard Ehrenborg and Margie Readdy, About Your Two Conjectures, But One Is FAMOUS, While The Other Is FALSE</a>; <a href="/A072187/a072187.pdf">Local copy</a> [PDF file only, no active links]
%F A072187 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
%e A072187 a(3)=1 since among the four involutions of length 3 (123, 213, 321, 132), only one is up-down (132).
%Y A072187 Cf. A000085, A000111.
%K A072187 nonn
%O A072187 0,5
%A A072187 _Doron Zeilberger_, Jul 01 2002; more terms, Dec 09 2003
%E A072187 More terms from _Vladeta Jovovic_, May 16 2007
%E A072187 a(0)=1 prepended by _Alois P. Heinz_, Aug 07 2018