A274052 Number of factor-free Dyck words with slope 5/2 and length 7n.

1, 3, 13, 94, 810, 7667, 76998, 805560, 8684533, 95800850, 1076159466, 12268026894, 141565916433, 1650395185407, 19409211522550, 229984643863260, 2743097412254490, 32907239462485422, 396793477697214450, 4806417317271974580, 58460150525944945840, 713685698665966837135, 8742060290902752902340

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 (2n,5n) that stay below the line y = 5/2x and also do not contain a proper sub-path of smaller size.

Links

Table of n, a(n) for n=0..22.

Cyril Banderier, Michael Wallner, Lattice paths of slope 2/5, 2015 Proceedings of the Twelfth Workshop on Analytic Algorithmics and Combinatorics (ANALCO).

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.

Daniel Birmajer, Juan B. Gil, Michael D. Weiner, On factor-free Dyck words with half-integer slope, arXiv:1804.11244 [math.CO], 2018.

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(7*n, 2*n)*x^n/n ). Then A(x) = ( x/series reversion of x*E(x) )^(1/7) = 1 + 3*x + 13*x^2 + 94*x^3 + .... Equivalently, [x^n]( A(x)^(7*n) ) = binomial(7*n, 2*n) for n = 0,1,2,.... - Peter Bala, Jan 01 2020

Example

a(2) = 13 since there are 13 lattice paths (allowing only north and east steps) starting at (0,0) and ending at (4,10) that stay below the line y=5/2x and also do not contain a proper sub-path of small size; e.g., EEENNNENNNNNNN is a factor-free Dyck word but ENNEENENNNNNNN contains the factor ENENNNN.

Crossrefs

Factor-free Dyck words: A005807 (slope 3/2), A274244 (slope 7/2), A274256 (slope 9/2), A274257 (slope 4/3), A274259 (slope 7/3).

Cf. A293946, A322631.

Sequence in context: A257661 A292501 A243243 * A305207 A183283 A006898

Adjacent sequences: A274049 A274050 A274051 * A274053 A274054 A274055

Keyword

nonn

Author

Michael D. Weiner, Jun 08 2016

Status

approved