cp's OEIS Frontend

A274259 Number of factor-free Dyck words with slope 7/3 and length 10n.

%I A274259 #22 Dec 17 2023 16:41:44
%S A274259 1,12,570,44689,4223479,441010458,49014411306,5685822210429,
%T A274259 680500195656621,83406972284096638,10416465145620729162,
%U A274259 1320749077779826216029,169570747575202480367168,22000830732097549119672094,2880094468241888675318895339,379941591968957300338548388051,50458777676743899501139029335858
%N A274259 Number of factor-free Dyck words with slope 7/3 and length 10n.
%C A274259 a(n) is the number of lattice paths (allowing only north and east steps) starting at (0,0) and ending at (3n,7n) that stay below the line y=7/3x and also do not contain a proper subpath of smaller size.
%H A274259 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 A274259 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 A274259 Conjectural o.g.f.: Let E(x) = exp( Sum_{n >= 1} binomial(10*n, 3*n)*x^n/n ). Then A(x) = ( x/series reversion of x*E(x) )^(1/10) = 1 + 12*x + 570*x^2 + 44689*x^3 + ... . Equivalently, [x^n]( A(x)^(10*n) ) = binomial(10*n, 3*n) for n = 0,1,2,... . - _Peter Bala_, Jan 03 2020
%e A274259 a(2) = 570 since there are 570 lattice paths (allowing only north and east steps) starting at (0,0) and ending at (6,14) that stay below the line y=7/3x and also do not contain a proper subpath of small size; e.g., ENNENENNNENNENNNENNN is a factor-free Dyck word but ENNENNENNEENNNNNENNN contains the factor ENNEENNNNN.
%Y A274259 Factor-free Dyck words: A005807 (slope 3/2), A274052 (slope 5/2), A274244 (slope 7/2), A274256 (slope 9/2), A274257 (slope 4/3), A274258 (slope 5/3).
%K A274259 nonn
%O A274259 0,2
%A A274259 _Michael D. Weiner_, Jun 16 2016