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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 (list; graph; refs; listen; history; text; internal format)
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 n-paths (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 (n-1)-paths. A fixed point occurs when a vertex of a Dyck k-path 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.: [1-sqrt(1-4x)]/x - 1/(1-x). - 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(n-1) +(9*n-13)*a(n-2) +2*(-2*n+5)*a(n-3)=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

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Last modified April 20 22:29 EDT 2014. Contains 240823 sequences.