

A131428


2*C(n)1, where C(n)=A000108(n) are the Catalan numbers.


8



1, 1, 3, 9, 27, 83, 263, 857, 2859, 9723, 33591, 117571, 416023, 1485799, 5348879, 19389689, 70715339, 259289579, 955277399, 3534526379, 13128240839, 48932534039, 182965127279, 686119227299, 2579808294647, 9723892802903
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OFFSET

0,3


COMMENTS

Starting (1, 3, 9, 27, 83,...), = row sums of triangle A136522.  Gary W. Adamson, Jan 02 2008
Hankel transform is A171552.  Paul Barry, Dec 11 2009
Apparently, for n>=1, the maximum peak height minus the maximum valley height summed over all Dyck npaths (with max valley height deemed zero if no valleys).  David Scambler, Oct 05 2012
Apparently for n>1 the number of fixed points in all Dyck (n1)paths. A fixed point occurs when a vertex of a Dyck kpath is also a vertex of the path U^kD^k.  David Scambler, May 01 2013


LINKS

Table of n, a(n) for n=0..25.


FORMULA

Right border of triangle A131429.
a(n) = 2*binom(2n,n)/(n+1)  1. G.f.: [1sqrt(14x)]/x  1/(1x).  Emeric Deutsch, Jul 25 2007
(1, 3, 9, 27, 83,...) = row sums of A118976.  Gary W. Adamson, Aug 31 2007
Row sums of triangle A131428 starting (1, 3, 9, 27, 83,...).  Gary W. Adamson, Aug 31 2007
Starting with offset 1 = Narayana transform (A001263) of [1,2,2,2,...].  Gary W. Adamson, Jul 29 2011
Conjecture: (n+1)*a(n) +2*(3*n+1)*a(n1) +(9*n13)*a(n2) +2*(2*n+5)*a(n3)=0.  R. J. Mathar, Nov 30 2012


EXAMPLE

a(3) = 9 = 2*C(3)  1 = 2*5  1; where C refers to the Catalan numbers, A000108.


MAPLE

seq(2*binomial(2*n, n)/(n+1)1, n=0..25); # Emeric Deutsch, Jul 25 2007


MATHEMATICA

2CatalanNumber[Range[0, 25]]1 (* Harvey P. Dale, Apr 17 2011 *)


CROSSREFS

Cf. A000108, A131427, A131429.
Cf. A131428, A118976.
Cf. A136522.
Sequence in context: A237272 A192909 A171155 * A099787 A176826 A146786
Adjacent sequences: A131425 A131426 A131427 * A131429 A131430 A131431


KEYWORD

nonn


AUTHOR

Gary W. Adamson, Jul 10 2007


EXTENSIONS

More terms from Emeric Deutsch, Jul 25 2007


STATUS

approved



