editing

approved

D-finite with recurrence 6*(n+1)*a(n) +2*(-25*n+11)*a(n-1) +(131*n-229)*a(n-2) +2*(-92*n+261)*a(n-3) +2*(81*n-311)*a(n-4) +(-91*n+439)*a(n-5) +(31*n-183)*a(n-6) +5*(-n+7)*a(n-7)=0. - R. J. Mathar, Jul 26 2022

approved

editing

proposed

approved

editing

proposed

Number of skew Dyck paths of semilength n with no base pyramids. 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.

Number of skew Dyck paths of semilength n with no base pyramids.

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.

proposed

editing

editing

proposed

a(n)=A129165(n,0).

G. C. Greubel, <a href="/A129166/b129166.txt">Table of n, a(n) for n = 0..1000</a>

a(n) = A129165(n,0).

G.f.=: (1-z)[*(3-3z3*z-sqrt(1-6z6*z+5z5*z^2)])/[(2-(1-z)[*(1-z-sqrt(1-6z6*z+5z5*z^2)]])).

(PARI) z='z+O('z^50); Vec((1-z)*(3-3*z-sqrt(1-6*z+5*z^2))/(2-(1-z)*(1-z-sqrt(1-6*z+5*z^2)))) \\ G. C. Greubel, Mar 20 2017

approved

editing

editing

approved

a(n) ~ 82*5^(n+1/2)/(289*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 20 2014

CoefficientList[Series[(1-x)*(3-3*x-Sqrt[1-6*x+5*x^2])/(2-(1-x)*(1-x-Sqrt[1-6*x+5*x^2])), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)

approved

editing

_Emeric Deutsch (deutsch(AT)duke.poly.edu), _, Apr 04 2007