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

0,2

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 sub-path of smaller size.

Links

Table of n, a(n) for n=0..16.

Daniel Birmajer, Juan B. Gil, 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.

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

Example

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 sub-path of small size; e.g., ENNENENNNENNENNNENNN is a factor-free Dyck word but ENNENNENNEENNNNNENNN contains the factor ENNEENNNNN.

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).

Sequence in context: A133415 A192602 A227143 * A192603 A192604 A192605

Adjacent sequences: A274256 A274257 A274258 * A274260 A274261 A274262

Keyword

nonn

Author

Michael D. Weiner, Jun 16 2016

Status

approved