[seqfan] (no subject)

[Richard Mathar]

~s more Dyck Paths

On the bottom of page 282 of 

Shu-Chung Liu, Jun Ma, Yeong-Nan Yeh, "Dyck Paths with Peak-
and Valley-Avoiding Sets", Stud. Appl Math. 121 (3) (2008) 263-289
http://dx.doi.org/10.1111/j.1467-9590.2008.00415.x

we read

"Remark 1. The sequences formed by the coefficient of D_{bar2s,bar2s)(1,z)
and D_{bar0,3bar1,a)(1,z) are not in Sloane. The coefficients of
... which is the sequence A025241 in [4].."
[4] N. J. A. Sloane, "The On-line.." available at...

To quote my former thesis supervisor (in translation): "this has gotta change.."

Proposal: some volunteer who understands the notation
submits what appear to be

gf := (1-sqrt(1-4*x+4*x^2-4*x^3))/2/(1-x+x^2)/x ;
1,1,1,2,5,12,29,73,190,505,1363,3727,10306,28771,80975,229512,654545,1876899,5408142,15650939,45470545,132573406,387775229,1137575084,3346189045,9867291817,29163523978,86377998093,256343194011,762144206268,2269845327797,6770976136649

gf := (1+x-2*x^2-sqrt(1-2*x-3*x^2+4*x^4))/2/(1-x)/x ;
1,1,2,3,6,12,26,59,138,332,814,2028,5118,13054,33598,87143,227542,597640,1577866,4185108,11146570,29798682,79932298,215072896,580327122,1569942098,4257254850,11569980794,31508150890,85968266198,234975421554,643317390627,1764010289514

gf := (1-x-sqrt(1-2*x-3*x^2+4*x^4))/2/(1+x)/x^2 ;
1,0,1,2,4,10,23,56,138,344,870,2220,5716,14828,38717,101682,268416,711810,1895432,5066030,13586082,36547534,98593064,266661162,722953814,1964358938,5348367006,14589803090,39870312218,109136843138,299205125935,821487772952,2258550716370

gf :=  (1-x+x^2-sqrt(1-2*x-x^2-2*x^3+x^4))/2/x^2 ;
1,1,1,2,4,8,17,37,82,185,423,978,2283,5373,12735,30372,72832,175502,424748,1032004,2516347,6155441,15101701,37150472,91618049,226460893,560954047,1392251012,3461824644,8622571758,21511212261,53745962199,134474581374,336908488839

gf :=  (1-2*x+2*x^2-sqrt(1-4*x+4*x^2-4*x^3))/2/x^2 ;
1,1,2,4,9,22,56,146,388,1048,2869,7942,22192,62510,177308,506008,1451866,4185788,12119696,35227748,102753800,300672368,882373261,2596389190,7658677856,22642421206,67081765932,199128719896,592179010350,1764044315540,5263275015120

Richard Mathar