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A274258
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Number of factor-free Dyck words with slope 5/3 and length 8n.
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3
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1, 7, 133, 4140, 154938, 6398717, 281086555, 12882897819, 609038885805, 29481041746958, 1453894927584477, 72789271870852237, 3689808842747726368, 189006099916444293090, 9768094831949586349262, 508712466332195692590121, 26670630123516854616641671, 1406503552584980596900001922, 74559627811441047591493767590
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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 (3n,5n) that stay below the line y=5/3x 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(8*n, 3*n)*x^n/n ). Then A(x) = ( x/series reversion of x*E(x) )^(1/8) = 1 + 7*x + 133*x^2 + 4140*x^3 + .... Equivalently, [x^n]( A(x)^(8*n) ) = binomial(8*n, 3*n) for n = 0,1,2,.... - Peter Bala, Jan 01 2020
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EXAMPLE
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a(2) = 133 since there are 133 lattice paths (allowing only north and east steps) starting at (0,0) and ending at (6,10) that stay below the line y=5/3x and also do not contain a proper sub-path of small size; e.g., ENEEEENNNNENNNNN is a factor-free Dyck word but ENEENNENNNEENNNN contains the factor EENNENNN.
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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), A274256 (slope 9/2), A274257 (slope 4/3), A274259 (slope 7/3).
Sequence in context: A110111 A245318 A274788 * A251577 A082164 A229464
Adjacent sequences: A274255 A274256 A274257 * A274259 A274260 A274261
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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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