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A128714 Number of skew Dyck paths of semilength n ending with a left step.

%I A128714 #21 Jul 26 2022 15:53:47
%S A128714 0,0,1,4,15,58,232,954,4010,17156,74469,327168,1452075,6501156,
%T A128714 29326743,133166064,608188737,2791992736,12876049123,59626721244,
%U A128714 277150709717,1292583258866,6046985696778,28369001791034,133436435891480
%N A128714 Number of skew Dyck paths of semilength n ending with a left step.
%C A128714 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.
%C A128714 Number of skew Dyck paths of semilength n and ending with a down step is A033321(n).
%H A128714 G. C. Greubel, <a href="/A128714/b128714.txt">Table of n, a(n) for n = 0..1000</a>
%H A128714 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 A128714 G.f.: (1 - 3z - sqrt(1-6z+5z^2))/(1 + z + sqrt(1-6z+5z^2)).
%F A128714 G.f.: z(g-1)/(1-zg), where g = 1 + zg^2 + z(g-1) = (1 - z - sqrt(1-6z+5z^2))(2z).
%F A128714 a(n) ~ 2*5^(n+1/2)/(9*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 20 2014
%F A128714 D-finite with recurrence 2*(n+1)*a(n) + (-13*n+7)*a(n-1) + 2*(8*n-17)*a(n-2) + 5*(-n+3)*a(n-3) = 0. - _R. J. Mathar_, Jul 14 2016
%e A128714 a(3)=4 because we have UDUUDL, UUDUDL, UUUDDL and UUUDLL.
%p A128714 G:=(1-3*z-sqrt(1-6*z+5*z^2))/(1+z+sqrt(1-6*z+5*z^2)): Gser:=series(G,z=0,30): seq(coeff(Gser,z,n),n=0..27);
%t A128714 CoefficientList[Series[(1-3*x-Sqrt[1-6*x+5*x^2])/(1+x+Sqrt[1-6*x+5*x^2]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 20 2014 *)
%o A128714 (PARI) concat([0,0],Vec((1-3*z-sqrt(1-6*z+5*z^2))/(1+z+sqrt(1-6*z+5*z^2)) + O(z^50))) \\ _G. C. Greubel_, Jan 31 2017
%Y A128714 Cf. A033321.
%K A128714 nonn
%O A128714 0,4
%A A128714 _Emeric Deutsch_, Mar 30 2007