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

1, 12, 570, 44689, 4223479, 441010458, 49014411306, 5685822210429, 680500195656621, 83406972284096638, 10416465145620729162, 1320749077779826216029, 169570747575202480367168, 22000830732097549119672094, 2880094468241888675318895339, 379941591968957300338548388051, 50458777676743899501139029335858

Offset: 0

Author

Michael D. Weiner, Jun 16 2016

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

• Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, On rational Dyck paths and the enumeration of factor-free Dyck words, arXiv:1606.02183 [math.CO], 2016.
• P. Duchon, On the enumeration and generation of generalized Dyck words, Discrete Mathematics, 225 (2000), 121-135.

Crossrefs

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).

Formula

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