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%I #31 Dec 18 2023 00:43:47
%S 1,5,52,880,17856,399296,9491008,235274240,6014201600,157387037696,
%T 4195621863424,113534211297280,3110485641494528,86107512380129280,
%U 2404899661362184192,67680890349732102144,1917436905101367443456,54640222663002565640192,1565130555077611323392000,45039415225401829826232320
%N Number of factor-free Dyck words with slope 4/3 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 (3n,4n) that stay below the line y=4/3x 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 Daniel Birmajer, Juan B. Gil and Michael D. Weiner, <a href="https://doi.org/10.1016/j.dam.2018.02.020">On rational Dyck paths and the enumeration of factor-free Dyck words</a>, Discrete Applied Mathematics, 244 (2018), 36-43.
%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 G.f. satisfies: 0 = x*f^6 + x*(x-1)*f^5 - x^2*(x+1)*f^4 - x*(x-3)*(x+1)^2*f^3 + x*(x+1)^3*f^2 - (x+1)^4*f + (x+1)^5. - _Michael D. Weiner_, Jan 14 2019
%F Conjectural o.g.f.: Let E(x) = exp( Sum_{n >= 1} binomial(7*n, 3*n)*x^n/n ). Then A(x) = ( x/series reversion of x*E(x) )^(1/7) = 1 + 5*x + 52*x^2 + 880*x^3 + .... Equivalently, [x^n]( A(x)^(7*n) ) = binomial(7*n, 3*n) for n = 0,1,2,.... - _Peter Bala_, Jan 01 2020
%e a(2) = 52 since there are 52 lattice paths (allowing only north and east steps) starting at (0,0) and ending at (6,8) that stay below the line y=4/3x and also do not contain a proper subpath of small size; e.g., EENEENENNENNNN is a factor-free Dyck word but ENEEENNENNENNN contains the factor EENNENN.
%t m = 20; f[_] = 0;
%t Do[f[x_] = (1/(x+1)^4)(-(x^2 (x+1) f[x]^4) + x f[x]^6 + (x-1) x f[x]^5 - (x - 3) x (x+1)^2 f[x]^3 + x (x+1)^3 f[x]^2 + (x+1)^5) + O[x]^m, {m}];
%t CoefficientList[f[x], x] (* _Jean-François Alcover_, Sep 28 2019 *)
%Y Cf. A005807 (slope 3/2), A274052 (slope 5/2), A274244 (slope 7/2), A274256 (slope 9/2), A274258 (slope 5/3), A274259 (slope 7/3).
%K nonn
%O 0,2
%A _Michael D. Weiner_, Jun 16 2016
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