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.
LINKS
Robert Israel, Table of n, a(n) for n = 0..1432
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203
FORMULA
a(n) = Sum_{k>=0} k*A128738(n,k).
G.f.: (2zg - g - z + 1)/(3zg - z - 1), where g = 1 + zg^2 + z(g-1) = (1 - z - sqrt(1 - 6z + 5z^2))/(2z).
Conjecture: +2*(n+1)*(n^2-5*n+5)*a(n) +(-13*n^3+60*n^2-52*n+15)*a(n-1) +2*(8*n^3-44*n^2+63*n-20)*a(n-2) -5*(n-3)*(n^2-3*n+1)*a(n-3)=0. - R. J. Mathar, Jun 17 2016
Conjecture verified using the differential equation (-12*z^2+10*z+10)*y(z)+(-25*z^4-10*z^3+37*z^2-6*z)*y'(z)+(-30*z^5+56*z^4-18*z^3-2*z^2)*y''(z)+(-5*z^6+16*z^5-13*z^4+2*z^3)*y'''(z)+6*z^2=0 satisfied by the G.f.. - Robert Israel, Dec 25 2017
EXAMPLE
a(3)=6 because each of the paths UDUUDD, UUDDUD, UUDUDD, UUUDDL contains one DD, the path UUUDDD contains 2 DD's and the paths UDUDUD, UDUUDL, UUUDLD, UUDUDL and UUUDLL contain no DD's.
MAPLE
g:=(1-z-sqrt(1-6*z+5*z^2))/2/z: ser:=series((2*z*g-g-z+1)/(3*z*g-z-1), z=0, 30): seq(coeff(ser, z, n), n=0..27);
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Mar 31 2007
STATUS
approved