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%I
%S 1,1,2,5,15,51,191,772,3320,15032,71084,348889,1768483,9220655,
%T 49286863,269346822,1501400222,8519796094,49133373040,287544553912,
%U 1705548000296,10241669069576,62201517142632,381749896129920,2365758616886432,14793705539872672
%N Number of set partitions of {1, ..., n} that avoid enhanced 3-crossings (or enhanced 3-nestings).
%C Also the number of 2-regular 3-noncrossing partitions. There is a bijection from 2-regular 3-noncrossing partitions of n to enhanced partition of n-1. - Jing Qin (qj(AT)cfc.nankai.edu.cn), Oct 30 2007
%C It appears that this is the number of sequences of length n, starting with a(1) = 1 and 1 <= a(2) <= 2, with 1 <= a(n) <= max(a(n-1),a(n-2)) + 1 for n > 2. - _Franklin T. Adams-Watters_, May 27 2008
%D Sophie Burrill, Sergi Elizalde, Marni Mishna and Lily Yen, A generating tree approach to k-nonnesting partitions and permutations, Arxiv preprint arXiv:1108.5615, 2011
%H Alois P. Heinz, <a href="/A108307/b108307.txt">Table of n, a(n) for n = 0..350</a>
%H M. Bousquet-Melou and G. Xin, <a href="http://fr.arXiv.org/abs/math.CO/0506551">On partitions avoiding 3-crossings</a>, math.CO/0506551.
%H Chen, W., Deng, E., Du, R., Stanley, R. P. and Yan, C., <a href="http://fr.arXiv.org/abs/math.CO/0501230">Crossings and nestings of matchings and partitions</a>, math.CO/0501230
%H Emma Y. Jin, Jing Qin and Christian M. Reidys, <a href="http://www.combinatorics.cn/cbpc/paper.html">On 2-regular k-noncrossing partitions</a>, math.CO/07105014.
%F Recurrence: 8*(n+3)*(n+1)*a(n)+(7*n^2+53*n+88)*a(n+1)-(n+8)*(n+7)*a(n+2)=0 - Jing Qin (qj(AT)cfc.nankai.edu.cn), Oct 26 2007
%F G.f.: -(6*x^4-15*x^3-7*x^2-11*x-1)/(6*x^5)+(224*x^3-60*x^2+45*x+5) * hypergeom([1/3, 2/3],[2],27*x^2/(1-2*x)^3) / (30*x^5*(2*x-1))+(32*x^2+64*x+5) * hypergeom([2/3, 4/3],[3],27*x^2/(1-2*x)^3)/(5*x^3*(2*x-1)^2). - Mark van Hoeij, Oct 24 2011
%e There are 52 partitions of 5 elements, but a(5)=51 because the partition (1,5)(2,4)(3) has an enhanced 3-nesting
%p a:= proc (n) option remember; if n<=1 then 1 elif n=2 then 2 else (8*(n+1) *(n-1) *a(n-2)+ (7*(n-2)^2 +53*(n-2) +88) *a(n-1))/(n+6)/(n+5) fi end: seq (a(n), n=0..20); # _Alois P. Heinz_, Sep 05 2008
%Y Cf. A124303, A073525, A007317.
%Y Cf. A000110, A000108.
%K easy,nonn
%O 0,3
%A _Mireille Bousquet-Mélou_, Jun 29 2005
%E Edited by _N. J. A. Sloane_ at the suggestion of _Franklin T. Adams-Watters_, Apr 27 2008
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