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

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 A351342 A308520

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