A128723 Number of skew Dyck paths of semilength n having no peaks at level 1.
1, 0, 2, 6, 22, 84, 334, 1368, 5734, 24480, 106086, 465462, 2063658, 9231084, 41610162, 188820726, 861891478, 3954732384, 18230522422, 84390187986, 392120098258, 1828220666844, 8550445900442, 40103716079436
Offset: 0
Keywords
Examples
a(3)=6 because we have UUDUDD, UUUDDD, UUUDLD, UUDUDL, UUUDDL and UUUDLL.
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- E. Deutsch, E. Munarini and S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.
- Helmut Prodinger, Skew Dyck paths having no peaks at level 1, arXiv:2201.00640 [math.CO], 2022.
Programs
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Maple
G:=(3-3*z-sqrt(1-6*z+5*z^2))/(1+3*z+sqrt(1-6*z+5*z^2)): Gser:=series(G,z=0,30): seq(coeff(Gser,z,n),n=0..27);
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Mathematica
CoefficientList[Series[(3-3*x-Sqrt[1-6*x+5*x^2])/(1+3*x+Sqrt[1-6*x+5*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *) -
PARI
my(z='x+O('z^50)); Vec((3-3*z-sqrt(1-6*z+5*z^2)) /(1+3*z +sqrt(1-6*z+5*z^2))) \\ G. C. Greubel, Mar 19 2017
Formula
a(n) = A128722(n,0).
a(n) = 2*A117641(n) for n>=1.
G.f.: (3-3*z-sqrt(1-6*z+5*z^2))/(1+3*z+sqrt(1-6*z+5*z^2)).
a(n) ~ 5^(n+3/2)/(16*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 20 2014
Conjecture: +3*(n+1)*a(n) +(-17*n+10)*a(n-1) +9*(n-3)*a(n-2) +5*(n-2)*a(n-3)=0. - R. J. Mathar, Jun 17 2016
Comments