OFFSET
0,4
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. A primitive non-Dyck factor is a subpath of the form UPD, P being a skew Dyck path with at least one L step, or of the form UPL, P being any nonempty skew Dyck path.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203
FORMULA
a(n) = Sum_{k=0,..,n} k*A129157(n,k).
G.f.: (1-5*z+3*(1-z)*sqrt(1-4*z)-3*sqrt(1-6*z+5*z^2) - sqrt((1-4*z)*(1-6*z+5*z^2)))/(1+z+sqrt(1-6*z+5*z^2))^2.
a(n) ~ (3*sqrt(5)+5) * 5^(1+n) / (36*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 20 2014
EXAMPLE
a(2)=1 because in all skew Dyck paths of semilength 3, namely UDUD, UUDD and (UUDL), we have altogether 1 primitive non-Dyck factor (shown between parentheses).
MAPLE
G:=(1-5*z+3*(1-z)*sqrt(1-4*z)-3*sqrt(1-6*z+5*z^2)-sqrt((1-4*z)*(1-6*z+5*z^2)))/(1+z+sqrt(1-6*z+5*z^2))^2: Gser:=series(G, z=0, 32): seq(coeff(Gser, z, n), n=0..27);
MATHEMATICA
CoefficientList[Series[(1-5*x+3*(1-x)*Sqrt[1-4*x]-3*Sqrt[1-6*x+5*x^2]-Sqrt[(1-4*x)*(1-6*x+5*x^2)])/(1+x+Sqrt[1-6*x+5*x^2])^2, {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)
PROG
(PARI) z='z+O('z^25); concat([0, 0], Vec((1-5*z+3*(1-z)*sqrt(1-4*z)-3*sqrt(1-6*z+5*z^2) - sqrt((1-4*z)*(1-6*z+5*z^2))) /(1+z+ sqrt(1-6*z+5*z^2) )^2)) \\ G. C. Greubel, Feb 09 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Apr 02 2007
STATUS
approved