

A071718


Expansion of (1+x^2*C)*C^3, where C = (1(14*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.


4



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
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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 nelement set which either do not contain {n1,n} as a block, or which do not contain the block {n} whenever 1 and n1 are in the same block. a(n) is the cardinality of X_{n+2}. For example, a(1)=3 counts 123, 132, 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/(n1))*binomial(2n,n2).
(n+3)*a(n) + 2*(2*n3)*a(n1) + 2*(n+1)*a(n2) + 4*(2*n5)*a(n3) = 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



