OFFSET
0,3
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 and Vincenzo Librandi, Table of n, a(n) for n = 0..1000 (terms 0..300 from Vincenzo Librandi)
E. Deutsch, E. Munarini and S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.
Emeric Deutsch, Emanuele Munarini and Simone Rinaldi, Skew Dyck paths, area, and superdiagonal bargraphs, Journal of Statistical Planning and Inference, Vol. 140, Issue 6, June 2010, pp. 1550-1562, Table 1.
FORMULA
a(n) = Sum_{k=0,..,n^2} k*A129172(n,k).
G.f.: (1+z)*(1-3*z-sqrt(1-6*z+5*z^2))/(2*z*(1-5*z)).
(n+1)(n-2)a(n)-(11n^2-20n-6)a(n-1)+5(7n^2-19n+7)a(n-2)-25(n-1)(n-3)a(n-3) = 0.
a(n) ~ 6*5^(n-1)*(1-sqrt(5)/sqrt(Pi*n)) . - Vaclav Kotesovec, Oct 20 2012
EXAMPLE
a(2)=9 because the areas below the skew Dyck paths UDUD, UUDD and UUDL are 2, 4 and 3, respectively.
MAPLE
a[0]:=1: a[1]:=1: a[2]:=9: for n from 3 to 25 do a[n]:=((11*n^2-20*n-6)*a[n-1]-5*(7*n^2-19*n+7)*a[n-2]+25*(n-1)*(n-3)*a[n-3])/(n+1)/(n-2) od: seq(a[n], n=0..25);
MATHEMATICA
CoefficientList[Series[(1+x)*(1-3*x-Sqrt[1-6*x+5*x^2])/(2*x*(1-5*x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)
PROG
(PARI) z='z +O('z^25); concat([0], Vec((1+z)*(1-3*z-sqrt(1-6*z+5*z^2))/(2*z*(1-5*z)))) \\ G. C. Greubel, Feb 10 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Apr 09 2007
STATUS
approved