A129160 Sum of the semi-abscissae of the first returns to the axis over all skew Dyck paths of semilength n.
1, 4, 18, 82, 378, 1760, 8262, 39044, 185526, 885596, 4243590, 20400954, 98353278, 475322352, 2302064010, 11170370850, 54293503770, 264290420540, 1288257980310, 6287181414470, 30717958762350, 150234512678480, 735446569221810, 3603330368706640, 17668505697688098, 86698739895529300
Offset: 1
Keywords
Examples
a(2)=4 because UDUD, UUDD and UUDL yield 1+2+1=4.
Links
- G. C. Greubel, Table of n, a(n) for n = 1..1000 (terms 1..200 from Vincenzo Librandi)
- E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203
Programs
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Maple
G:=z-1+(1-3*z+2*z^2)/sqrt(1-6*z+5*z^2): Gser:=series(G,z=0,30): seq(coeff(Gser,z,n),n=1..27);
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Mathematica
CoefficientList[Series[(1/x) (x - 1 + (1 - 3*x + 2*x^2)/Sqrt[1 - 6*x + 5*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *) -
PARI
x='x+O('x^25); Vec(x-1+(1-3*x+2*x^2)/sqrt(1-6*x+5*x^2)) \\ G. C. Greubel, Feb 09 2017
Formula
a(n) = Sum_{k=1,..,n} k*A129159(n,k).
a(n) = 2*A128752(n) for n>=2.
G.f.: x-1+(1-3*x+2*x^2)/sqrt(1-6*x+5*x^2).
Recurrence: n*(3*n-1)*a(n) = 18*(n-1)*n*a(n-1) - 5*(n-3)*(3*n+2)*a(n-2) . - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 6*5^(n-3/2)/sqrt(Pi*n) . - Vaclav Kotesovec, Oct 20 2012
Extensions
Mathematica code corrected by Vincenzo Librandi, May 24 2013
Comments