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A274256
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Number of factor-free Dyck words with slope 9/2 and length 11n.
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6
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1, 5, 70, 1696, 49493, 1593861, 54591225, 1950653202, 71889214644, 2712628146949, 104277713515456, 4069334248174800, 160785480249706192, 6419443865094494044, 258585021917711797850, 10496205397574996367474, 428899108081734423242550, 17628723180468295514015268, 728347675604866545590505024
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OFFSET
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0,2
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COMMENTS
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a(n) is the number of lattice paths (allowing only north and east steps) starting at (0,0) and ending at (2n,9n) that stay below the line y=9/2x and also do not contain a proper sub-path of smaller size.
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LINKS
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Table of n, a(n) for n=0..18.
Daniel Birmajer, Juan B. Gil, Michael D. Weiner, On rational Dyck paths and the enumeration of factor-free Dyck words, arXiv:1606.02183 [math.CO], 2016.
P. Duchon, On the enumeration and generation of generalized Dyck words, Discrete Mathematics, 225 (2000), 121-135.
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FORMULA
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Conjectural o.g.f.: Let E(x) = exp( Sum_{n >= 1} binomial(11*n,2*n)*x^n/n ). Then A(x) = ( x/series reversion of x*E(x) )^(1/11) = 1 + 5*x + 70*x^2 + 1696*x^3 + .... Equivalently, [x^n]( A(x)^(11*n) ) = binomial(11*n, 2*n) for n = 0,1,2,.... - Peter Bala, Jan 01 2020
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EXAMPLE
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a(2) = 70 since there are 70 lattice paths (allowing only north and east steps) starting at (0,0) and ending at (4,18) that stay below the line y=9/2x and also do not contain a proper sub-path of small size; e.g., ENNENNENNNNNNENNNNNNNN is a factor-free Dyck word but ENEENENNNNNNNNNNNNNNNN contains the factor ENENNNNNNNN.
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CROSSREFS
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Factor-free Dyck words: A005807 (slope 3/2), A274052 (slope 5/2), A274244 (slope 7/2), A274257 (slope 4/3), A274258 (slope 5/3), A274259 (slope 7/3).
Sequence in context: A014231 A203528 A147629 * A280574 A302910 A174486
Adjacent sequences: A274253 A274254 A274255 * A274257 A274258 A274259
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KEYWORD
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nonn
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AUTHOR
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Michael D. Weiner, Jun 16 2016
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STATUS
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approved
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