A131428 a(n) = 2*C(n) - 1, where C(n) = A000108(n) are the Catalan numbers.

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, 36734706144303, 139067101832007

Offset: 0

Author

Gary W. Adamson, Jul 10 2007

Keywords

• nonn

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

Examples

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

Links

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

Crossrefs

Cf. A000108, A131427, A131429.

Cf. A131428, A118976, A136522.

Cf. A080233, A156644.

Programs

GAP

List([0..25], 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

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 *)

PARI

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

Sage

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

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

D-finite with recurrence (n+1)*a(n) +2*(-2*n+1)*a(n-1) +3*(-n+1)=0. - R. J. Mathar, Nov 22 2024

a(n) = Sum_{k=0..n} ( binomial(n,k) - binomial(n,k-1) )^2 = Sum_{k=0..n} A080233(n,k)^2 = Sum_{k=0..n} A156644(n,k)^2. - Seiichi Manyama, Mar 25 2025

Extensions

More terms from Emeric Deutsch, Jul 25 2007