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
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
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
KEYWORD
nonn
AUTHOR
Gary W. Adamson, Jul 10 2007
EXTENSIONS
More terms from Emeric Deutsch, Jul 25 2007
STATUS
approved