A071718 Expansion of (1+x^2*C)*C^3, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.

1, 3, 10, 32, 104, 345, 1166, 4004, 13936, 49062, 174420, 625328, 2258416, 8209045, 30008790, 110255100, 406923360, 1507973610, 5608843020, 20931740640, 78354322800, 294127079610, 1106939020044, 4175827174152, 15787544777504

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

Table of n, a(n) for n=0..24.

M. Bruce, M. Dougherty, M. Hlavacek, R. Kudo, I. Nicolas, A Decomposition of Parking Functions by Undesired Spaces, The Electronic Journal of Combinatorics 23(3), 2016.

H. Mühle, Two Posets of Noncrossing Partitions Coming From Undesired Parking Spaces, arXiv:1701.02109 [math.CO], 2017.

Formula

For n>1, a(n) = 3*A000245(n) + A000344(n) = (5/(n+3) + 9/(n-1))*binomial(2n,n-2).

(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

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

Sequence in context: A033505 A297067 A063782 * A261058 A306295 A356499

Adjacent sequences: A071715 A071716 A071717 * A071719 A071720 A071721

Keyword

nonn

Author

N. J. A. Sloane, Jun 06 2002

Status

approved