OFFSET
0,2
COMMENTS
a(n) = number of Dyck (n+3)-paths whose third from last upstep initiates a long ascent, n>=1. A long ascent is one consisting of 2 or more upsteps. For example, a(1)=3 counts UDuUUDDD, UDuUDUDD, UDuUDDUD (third from last upstep in small type). - David Callan, Dec 08 2004
For n>0 a(n)=number of Dyck (n+3)-paths whose 5th and 6th steps are DU. For example, a(1)=3 counts UDUUduDD, UUDUduDD, UUUDduDD. - David Scambler, Feb 14 2011
Let X_n be the set of all noncrossing set partitions of an n-element set which either do not contain {n-1,n} as a block, or which do not contain the block {n} whenever 1 and n-1 are in the same block. a(n) is the cardinality of X_{n+2}. For example, a(1)=3 counts 1|2|3, 13|2, 123. - Henri Mühle, Jan 10 2017
LINKS
Melody Bruce, Michael Dougherty, Max Hlavacek, Ryo Kudo, and Ian Nicolas, A Decomposition of Parking Functions by Undesired Spaces, The Electronic Journal of Combinatorics 23(3), 2016.
Henri Mühle, Two Posets of Noncrossing Partitions Coming From Undesired Parking Spaces, arXiv:1701.02109 [math.CO], 2017.
FORMULA
D-finite with recurrence (n+3)*a(n) + 2*(-2*n-3)*a(n-1) + 2*(-n+1)*a(n-2) + 4*(2*n-5)*a(n-3) = 0. - R. J. Mathar, Aug 25 2013
a(n) ~ 7 * 2^(2*n+1) / (n^(3/2) * sqrt(Pi)). - Amiram Eldar, Oct 01 2025
MATHEMATICA
{1, 3}~Join~Table[(5/(n + 3) + 9/(n - 1))*Binomial[2 n, n - 2], {n, 2, 24}] (* Michael De Vlieger, Jan 10 2017 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Jun 06 2002
STATUS
approved
