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A129166 Number of skew Dyck paths of semilength n with no base pyramids.

%I A129166 #13 Jul 26 2022 10:58:41
%S A129166 1,0,1,5,19,73,292,1203,5065,21697,94274,414514,1840981,8247011,
%T A129166 37220261,169079113,772489020,3547371679,16364309243,75799327800,
%U A129166 352402156770,1643878188646,7691841654538,36091803172733
%N A129166 Number of skew Dyck paths of semilength n with no base pyramids.
%C A129166 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.
%H A129166 G. C. Greubel, <a href="/A129166/b129166.txt">Table of n, a(n) for n = 0..1000</a>
%H A129166 E. Deutsch, E. Munarini, S. Rinaldi, <a href="http://dx.doi.org/10.1016/j.jspi.2010.01.015">Skew Dyck paths</a>, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203
%F A129166 a(n) = A129165(n,0).
%F A129166 G.f.: (1-z)*(3-3*z-sqrt(1-6*z+5*z^2))/(2-(1-z)*(1-z-sqrt(1-6*z+5*z^2))).
%F A129166 a(n) ~ 82*5^(n+1/2)/(289*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 20 2014
%F A129166 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
%e A129166 a(2)=1 because we have UUDL.
%p A129166 G:=(1-z)*(3-3*z-sqrt(1-6*z+5*z^2))/(2-(1-z)*(1-z-sqrt(1-6*z+5*z^2))): Gser:=series(G,z=0,30): seq(coeff(Gser,z,n),n=0..27);
%t A129166 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 *)
%o A129166 (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
%Y A129166 Cf. A129165.
%K A129166 nonn
%O A129166 0,4
%A A129166 _Emeric Deutsch_, Apr 04 2007