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Revision History for A274244 (Bold, blue-underlined text is an addition; faded, red-underlined text is a deletion.)

Showing entries 1-10 | older changes
Number of factor-free Dyck words with slope 7/2 and length 9n.
(history; published version)
#25 by Alois P. Heinz at Sun Dec 17 17:36:45 EST 2023
STATUS

proposed

approved

#24 by Stefano Spezia at Sun Dec 17 15:09:37 EST 2023
STATUS

editing

proposed

#23 by Stefano Spezia at Sun Dec 17 15:09:34 EST 2023
FORMULA

Conjectural o.g.f.: Let E(x) = exp( Sum_{n >= 1} binomial(9*n, 2*n)*x^n/n ). Then A(x) = ( x/series reversion of x*E(x) )^(1/9) = 1 + 4*x + 34*x^2 + 494*x^3 + ... . Equivalently, [x^n]( A(x)^(9*n) ) = binomial(9*n, 2*n) for n = 0,1,2,... . - Peter Bala, Jan 01 2020

STATUS

proposed

editing

#22 by Michel Marcus at Sun Dec 17 15:01:33 EST 2023
STATUS

editing

proposed

#21 by Michel Marcus at Sun Dec 17 15:01:31 EST 2023
LINKS

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.

STATUS

proposed

editing

#20 by Robert C. Lyons at Sun Dec 17 14:58:23 EST 2023
STATUS

editing

proposed

#19 by Robert C. Lyons at Sun Dec 17 14:58:21 EST 2023
COMMENTS

a(n) is the number of lattice paths (allowing only north and east steps) starting at (0,0) and ending at (2n,7n) that stay below the line y=7/2x and also do not contain a proper sub-path subpath of smaller size.

EXAMPLE

a(2) = 34 since there are 34 lattice paths (allowing only north and east steps) starting at (0,0) and ending at (4,14) that stay below the line y=7/2x and also do not contain a proper sub-path subpath of small size; e.g., EEENNNNENNNNNNNNNN is a factor-free Dyck word but EEENNENNNNNNNNNNNN contains the factor ENNENNNNN.

STATUS

approved

editing

#18 by Peter Luschny at Mon Jan 06 17:21:42 EST 2020
STATUS

proposed

approved

#17 by Michel Marcus at Mon Jan 06 01:01:58 EST 2020
STATUS

editing

proposed

#16 by Michel Marcus at Mon Jan 06 01:01:56 EST 2020
FORMULA

Conjectural o.g.f.: Let E(x) = exp( Sum_{n >= 1} binomial(9*n, 2*n)*x^n/n ). Then A(x) = ( x/series reversion of x*E(x) )^(1/9) = 1 + 4*x + 34*x^2 + 494*x^3 + .... Equivalently, [x^n]( A(x)^(9*n) ) = binomial(9*n, 2*n) for n = 0,1,2,.... - Peter Bala, Jan 01 2020

STATUS

proposed

editing