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A131428
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a(n) = 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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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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FORMULA
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a(n) = 2*binomial(2*n,n)/(n+1) - 1.
G.f.: (1-sqrt(1-4*x))/x - 1/(1-x). (End)
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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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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