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%I #40 Dec 18 2023 00:31:04
%S 1,3,13,94,810,7667,76998,805560,8684533,95800850,1076159466,
%T 12268026894,141565916433,1650395185407,19409211522550,
%U 229984643863260,2743097412254490,32907239462485422,396793477697214450,4806417317271974580,58460150525944945840,713685698665966837135,8742060290902752902340
%N Number of factor-free Dyck words with slope 5/2 and length 7n.
%C a(n) is the number of lattice paths (allowing only north and east steps) starting at (0,0) and ending at (2n,5n) that stay below the line y = 5/2x and also do not contain a proper subpath of smaller size.
%H Cyril Banderier and Michael Wallner, <a href="http://dx.doi.org/10.1137/1.9781611973761.10">Lattice paths of slope 2/5</a>, 2015 Proceedings of the Twelfth Workshop on Analytic Algorithmics and Combinatorics (ANALCO).
%H Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, <a href="http://arxiv.org/abs/1606.02183">On rational Dyck paths and the enumeration of factor-free Dyck words</a>, arXiv:1606.02183 [math.CO], 2016.
%H Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, <a href="https://arxiv.org/abs/1804.11244">On factor-free Dyck words with half-integer slope</a>, arXiv:1804.11244 [math.CO], 2018.
%H P. Duchon, <a href="http://dx.doi.org/10.1016/S0012-365X(00)00150-3">On the enumeration and generation of generalized Dyck words</a>, Discrete Mathematics, 225 (2000), 121-135.
%F Conjectural o.g.f.: Let E(x) = exp( Sum_{n >= 1} binomial(7*n, 2*n)*x^n/n ). Then A(x) = ( x/series reversion of x*E(x) )^(1/7) = 1 + 3*x + 13*x^2 + 94*x^3 + ... . Equivalently, [x^n]( A(x)^(7*n) ) = binomial(7*n, 2*n) for n = 0,1,2,... . - _Peter Bala_, Jan 01 2020
%e a(2) = 13 since there are 13 lattice paths (allowing only north and east steps) starting at (0,0) and ending at (4,10) that stay below the line y=5/2x and also do not contain a proper subpath of small size; e.g., EEENNNENNNNNNN is a factor-free Dyck word but ENNEENENNNNNNN contains the factor ENENNNN.
%Y Factor-free Dyck words: A005807 (slope 3/2), A274244 (slope 7/2), A274256 (slope 9/2), A274257 (slope 4/3), A274259 (slope 7/3).
%Y Cf. A293946, A322631.
%K nonn
%O 0,2
%A _Michael D. Weiner_, Jun 08 2016
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