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. A pyramid in a skew Dyck word (path) is a factor of the form u^h d^h, h being the height of the pyramid. A base pyramid is a pyramid starting on the x-axis.
a(n+1) is the number of 3-colored Motzkin paths of length n with no peaks at level 1. - José Luis Ramírez Ramírez, Mar 31 2013
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
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*A129165(n,k).
G.f.: (1 - 3*z - sqrt(1 - 6*z + 5*z^2))/(z*(3 - 3*z - sqrt(1 - 6*z + 5*z^2))).
Recurrence: 2*(n+1)*a(n) = (13*n-3)*a(n-1) - 4*(4*n-3)*a(n-2) + 5*(n-1)*a(n-3) . - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 5^(n+5/2)/(72*sqrt(Pi)*n^(3/2)) . - Vaclav Kotesovec, Oct 20 2012
EXAMPLE
a(2)=3 because in the paths (UD)(UD), (UUDD) and UUDL we have altogether 3 base pyramids (shown between parentheses).
MAPLE
G:=(1-3*z-sqrt(1-6*z+5*z^2))/z/(3-3*z-sqrt(1-6*z+5*z^2)): Gser:=series(G, z=0, 30): seq(coeff(Gser, z, n), n=0..27);
MATHEMATICA
CoefficientList[Series[(1-3*x-Sqrt[1-6*x+5*x^2])/(x*(3-3*x-Sqrt[1-6*x+5*x^2])), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)
PROG
(PARI) z='z+O('z^66); concat([0], Vec((1-3*z-sqrt(1-6*z+5*z^2))/z/(3-3*z-sqrt(1-6*z+5*z^2)))) \\ Joerg Arndt, Aug 27 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Apr 04 2007
STATUS
approved