%I A006789 M1462
%S A006789 1,1,2,5,14,43,143,509,1922,7651,31965,139685,636712,3020203,14878176,
%T A006789 75982829,401654560,2194564531,12377765239,71980880885,431114329728,
%U A006789 2656559925883,16825918195484,109439943234749,730365368850192
%N A006789 Bessel numbers: the number of nonoverlapping partitions of an n-set into 
               equivalence classes.
%C A006789 Comment from Lara Pudwell (Lara.Pudwell(AT)valpo.edu), Oct 23 2008 (Start):
%C A006789 A permutation p avoids a pattern q if it has no subsequence that is order-isomorphic 
               to q. For example, p avoids the pattern 132 if it has no subsequence 
               abc with a<c<b.
%C A006789 Barred pattern avoidance considers permutations that avoid a pattern 
               except in a special case. Given a barred pattern q, we may form two 
               patterns, q1 = the sequence of unbarred letters of q and q2 = the 
               sequence of all letters of q.
%C A006789 A permutation p avoids barred pattern q if every instance of q1 in p 
               is embedded in a copy of q2 in p. In other words, p avoids q1, except 
               in the special case that a copy of q1 is a subsequence of a copy 
               of q2.
%C A006789 For example, if q=5{bar 1}32{bar 4}, then q1=532 and q2 = 51324. p avoids 
               q if every for decreasing subsequence acd of length 3 in p, one can 
               find letters b and e so that the subsequence abcde of p has b<d<c<e<a. 
               (End)
%C A006789 Nonoverlapping means that the intervals associated with the minimum to 
               maximum integers of any two blocks of a partition do not overlap. 
               Instead, the intervals are disjoint or one contains another.
%C A006789 Apparently, also the number of permutations in S_n avoiding 2{bar 5}3{bar 
               1}4 (i.e. every occurrence of 234 is contained in an occurrence of 
               a 25314). - Lara Pudwell (Lara.Pudwell(AT)valpo.edu), Apr 25 2008
%C A006789 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 20 2008: 
               (Start)
%C A006789 Convolved with A153197 = A006789 shifted: (1, 2, 5, 14,...); equivalent 
               to row sums of triangle A153206 = (1, 2, 5, 14,...).
%C A006789 Equals inverse binomial transform of A153197 and INVERT transform of 
               A153197 prefaced with a 1.
%C A006789 Can be generated from the Hankel transform [1,1,1,...] through successive
%C A006789 iterative operations of: binomial transform, INVERT transform, binomial
%C A006789 transform, (repeat)...; or starting with INVERT transform. The operations
%C A006789 converge to a two sequence limit cycle of A006789 and its binomial transform, 
               A153197.
%C A006789 Shifts to (1, 2, 5, 14,...) with A006789 * A153197 prefaced with a 1; 
               i.e.
%C A006789 (1, 2, 5, 14, 43,...) = (1, 1, 2, 5, 14,...) * (1, 1, 2, 5, 15,...); 
               where A153197 = (1, 2, 5, 15, 51, 189, 748, 3128, 13731,...). (End)
%C A006789 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 21 2008: 
               (Start)
%C A006789 a(n) = term (1,1) of M^n, where M = an infinite Cartan-like matrix with 
               1's
%C A006789 the super- and subdiagonals (diagonals starting at (1,2) and (2,1)
%C A006789 respectively; and the main diagonal = (1,2,3,...). (End)
%D A006789 A. Claesson, Generalized pattern avoidance, Europ. J. Combin., 22 (2001), 
               961-971.
%D A006789 M. Klazar, Bell numbers, their relatives and algebraic differential equations, 
               J. Combin. Theory, A 102 (2003), 63-87.
%D A006789 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A006789 Lara Pudwell, <a href="http://faculty.valpo.edu/lpudwell/papers/pudwell_thesis.pdf">
               Enumeration Schemes for Pattern-Avoiding Words and Permutations</
               a>, Ph. D. Dissertation, Math. Dept., Rutgers University, May 2008.
%H A006789 M. Klazar, <a href="http://kam.mff.cuni.cz/~klazar/bell.ps">Bell numbers, 
               their relatives and algebraic differential equations</a>
%H A006789 A. Claesson and T. Mansour, <a href="http://arXiv.org/abs/math.CO/0107044">
               Permutations avoiding a pair of generalized patterns...</a>.
%H A006789 P. Flajolet and R. Schott, <a href="http://algo.inria.fr/flajolet/Publications/
               publist.html">Non-overlapping Partitions, Continued Fractions, Bessel 
               Functions and a Divergent Series</a> In European Journal of Combinatorics, 
               Vol. 11, 1990, pp. 412-432.
%H A006789 <a href="Sindx_Be.html#Bessel">Index entries for sequences related to 
               Bessel functions or polynomials</a>
%F A006789 G.f. 1/(1-x-x^2/(1-2x-x^2/(1-3x-x^2/...))) (a continued fraction).
%o A006789 (PARI) {a(n)=local(A); if(n<0, 0, A=O(x^0); for(i=0,n\2, A=subst((1+x)/
               (1-x^2*A),x,x/(1-x))); polcoeff(A,n))} /* Michael Somos Sep 22 2005 
               */
%o A006789 (PARI) {a(n)=local(m); if(n<0, 0, m=contfracpnqn(matrix(2, n\2, i, k, 
               if(i==1, -x^2, 1-(k+1)*x))); polcoeff(1/(1-x+m[2,1]/m[1,1])+x*O(x^n), 
               n))}
%Y A006789 Cf. A000110.
%Y A006789 A153197 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 20 2008]
%Y A006789 Sequence in context: A005425 A035349 A155888 this_sequence A098569 A137549 
               A014327
%Y A006789 Adjacent sequences: A006786 A006787 A006788 this_sequence A006790 A006791 
               A006792
%K A006789 nonn
%O A006789 0,3
%A A006789 N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
%E A006789 Edited by Michael Somos, Oct 06, 2003