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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. 3
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 (list; graph; refs; listen; history; text; internal format)
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 A134952

Adjacent sequences:  A071715 A071716 A071717 * A071719 A071720 A071721

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jun 06 2002

STATUS

approved

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Last modified October 17 06:08 EDT 2019. Contains 328106 sequences. (Running on oeis4.)