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A131428
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2*C(n)-1, where C(n)=A000108(n) are the Catalan numbers.
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8
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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
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OFFSET
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0,3
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COMMENTS
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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
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LINKS
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Table of n, a(n) for n=0..25.
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FORMULA
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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
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EXAMPLE
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a(3) = 9 = 2*C(3) - 1 = 2*5 - 1; where C refers to the Catalan numbers, A000108.
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MAPLE
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seq(2*binomial(2*n, n)/(n+1)-1, n=0..25); # Emeric Deutsch, Jul 25 2007
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MATHEMATICA
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2CatalanNumber[Range[0, 25]]-1 (* From Harvey P. Dale, Apr 17 2011 *)
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CROSSREFS
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Cf. A000108, A131427, A131429.
Cf. A131428, A118976.
Cf. A136522.
Sequence in context: A099786 A192909 A171155 * A099787 A176826 A146786
Adjacent sequences: A131425 A131426 A131427 * A131429 A131430 A131431
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KEYWORD
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nonn
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AUTHOR
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Gary W. Adamson, Jul 10 2007
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EXTENSIONS
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More terms from Emeric Deutsch, Jul 25 2007
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STATUS
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approved
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