OFFSET
1,2
COMMENTS
A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps.
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
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
EXAMPLE
a(2)=4 because UDUD, UUDD and UUDL yield 1+2+1=4.
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);
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Apr 03 2007
EXTENSIONS
Mathematica code corrected by Vincenzo Librandi, May 24 2013
STATUS
approved