0,3

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

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 (FrankTAW(AT)Netscape.net), May 27 2008

M. Bousquet-Melou and G. Xin, On partitions avoiding 3-crossings, math.CO/0506551.

Chen, W., Deng, E., Du, R., Stanley, R. P. and Yan, C., Crossings and nestings of matchings and partitions, math.CO/0501230

Emma Y. Jin, Jing Qin and Christian M. Reidys, On 2-regular k-noncrossing partitions, math.CO/07105014.

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

There are 52 partitions of 5 elements, but a(5)=51 because the partition (1,5)(2,4)(3) has an enhanced 3-nesting

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); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Sep 05 2008]

Cf. A124303 A073525 A007317.

Cf. A000110, A000108.

Sequence in context: A073525 A007317 A153197 this_sequence A117426 A001681 A053553

Adjacent sequences: A108304 A108305 A108306 this_sequence A108308 A108309 A108310

easy,nonn

Mireille Bousquet-Melou (bousquet(AT)labri.fr), Jun 29 2005

Edited by N. J. A. Sloane (njas(AT)research.att.com) at the suggestion of Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Apr 27 2008