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%I #20 Dec 17 2023 17:37:29
%S 1,7,133,4140,154938,6398717,281086555,12882897819,609038885805,
%T 29481041746958,1453894927584477,72789271870852237,
%U 3689808842747726368,189006099916444293090,9768094831949586349262,508712466332195692590121,26670630123516854616641671,1406503552584980596900001922,74559627811441047591493767590
%N Number of factor-free Dyck words with slope 5/3 and length 8n.
%C 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 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(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
%e 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 subpath of small size; e.g., ENEEEENNNNENNNNN is a factor-free Dyck word but ENEENNENNNEENNNN contains the factor EENNENNN.
%Y 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).
%K nonn
%O 0,2
%A _Michael D. Weiner_, Jun 16 2016
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