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%I #25 Dec 17 2023 17:36:45
%S 1,4,34,494,8615,165550,3380923,71999763,1580990725,35537491360,
%T 813691565184,18911247654404,444978958424224,10579389908116344,
%U 253756528273411250,6133110915783398175,149219383150626519874,3651756292682801022384,89830021324956206790496,2219945238901447637080235,55088272581138888326634644
%N Number of factor-free Dyck words with slope 7/2 and length 9n.
%C a(n) is the number of lattice paths (allowing only north and east steps) starting at (0,0) and ending at (2n,7n) that stay below the line y=7/2x and also do not contain a proper subpath of smaller size.
%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 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(9*n, 2*n)*x^n/n ). Then A(x) = ( x/series reversion of x*E(x) )^(1/9) = 1 + 4*x + 34*x^2 + 494*x^3 + ... . Equivalently, [x^n]( A(x)^(9*n) ) = binomial(9*n, 2*n) for n = 0,1,2,... . - _Peter Bala_, Jan 01 2020
%e a(2) = 34 since there are 34 lattice paths (allowing only north and east steps) starting at (0,0) and ending at (4,14) that stay below the line y=7/2x and also do not contain a proper subpath of small size; e.g., EEENNNNENNNNNNNNNN is a factor-free Dyck word but EEENNENNNNNNNNNNNN contains the factor ENNENNNNN.
%Y Factor-free Dyck words: A005807 (slope 3/2), A274052 (slope 5/2), A274256 (slope 9/2), A274257 (slope 4/3), A274259 (slope 7/3).
%Y Cf. A060941.
%K nonn
%O 0,2
%A _Michael D. Weiner_, Jun 15 2016
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