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 subpath of smaller size.
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.
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 subpath of small size; e.g., ENNENENNNENNENNNENNN is a factor-free Dyck word but ENNENNENNEENNNNNENNN contains the factor ENNEENNNNN.
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael D. Weiner, Jun 16 2016
STATUS
approved