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A131428 a(n) = 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

G. C. Greubel, Table of n, a(n) for n = 0..1000

FORMULA

Right border of triangle A131429.

From Emeric Deutsch, Jul 25 2007: (Start)

a(n) = 2*binomial(2*n,n)/(n+1) - 1.

G.f.: (1-sqrt(1-4*x))/x - 1/(1-x). (End)

(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 *)

PROG

(PARI) vector(25, n, n--; 2*binomial(2*n, n)/(n+1) - 1) \\ G. C. Greubel, Aug 12 2019

(MAGMA) [2*Catalan(n) -1: n in [0..25]]; // G. C. Greubel, Aug 12 2019

(Sage) [2*catalan_number(n) -1 for n in (0..25)] # G. C. Greubel, Aug 12 2019

(GAP) List([0..25], n-> 2*Binomial(2*n, n)/(n+1) - 1); # G. C. Greubel, Aug 12 2019

CROSSREFS

Cf. A000108, A131427, A131429.

Cf. A131428, A118976.

Cf. A136522.

Sequence in context: A192909 A047085 A171155 * A099787 A308520 A176826

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 July 23 14:44 EDT 2021. Contains 346259 sequences. (Running on oeis4.)