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A128750 Number of skew Dyck paths of semilength n having no ascents of length 1.

%I A128750 #21 Jul 23 2017 12:17:39
%S A128750 1,0,2,4,14,44,150,520,1850,6696,24602,91500,343846,1303572,4979822,
%T A128750 19150352,74075890,288022160,1125076210,4413061972,17375007294,
%U A128750 68641377980,272014578822,1081009104664,4307221752874,17203123381304
%N A128750 Number of skew Dyck paths of semilength n having no ascents of length 1.
%C A128750 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 a path is defined to be the number of its steps. An ascent in a path is a maximal sequence of consecutive U steps.
%C A128750 Hankel transform is 2^ceiling(n(n+1)/3). Binomial transform is A059278. - _Paul Barry_, Feb 11 2009
%H A128750 Vincenzo Librandi, <a href="/A128750/b128750.txt">Table of n, a(n) for n = 0..200</a>
%H A128750 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 A128750 a(n) = A128749(n,0).
%F A128750 G.f.: G = G(z) satisfies z(1 + z)G^2 - (1 - z^2)G + 1 - z = 0.
%F A128750 G.f.: 1/(1+x-x/(1-x-x/(1+x-x/(1-x-x/(1+x-x/(1-... (continued fraction). - _Paul Barry_, Feb 11 2009
%F A128750 From _Paul Barry_, Feb 11 2009: (Start)
%F A128750 G.f.: (1/(1+x))c(x/(1-x^2)) where c(x) is the g.f. of A000108;
%F A128750 G.f.: 1/(1-2x^2/(1-2x-x^2/(1-2x-2x^2/(1-x-2x^2/(1-2x-x^2/(1-2x-2x^2/(1-x-2x^2/(1-.... (continued fraction);
%F A128750 a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(floor((n+k)/2),k)*A000108(k).
%F A128750 (End)
%F A128750 Conjecture: (n+1)*a(n) +(-4*n+3)*a(n-1) +(-2*n-1)*a(n-2) +(4*n-11)*a(n-3) +(n-4)*a(n-4)=0. - _R. J. Mathar_, Nov 15 2012
%F A128750 a(n) ~ sqrt(5+3*sqrt(5)) * (2+sqrt(5))^n / (4 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Feb 08 2014
%e A128750 a(3)=4 because we have UUUDDD, UUUDLD, UUUDDL and UUUDLL.
%p A128750 G:=(1-z^2-sqrt((1-z^2)*(1-4*z-z^2)))/2/z/(1+z): Gser:=series(G,z=0,35): seq(coeff(Gser,z,n),n=0..30);
%t A128750 CoefficientList[Series[(1-x^2-Sqrt[(1-x^2)*(1-4*x-x^2)])/2/x/(1+x), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 08 2014 *)
%Y A128750 Cf. A128749.
%K A128750 nonn
%O A128750 0,3
%A A128750 _Emeric Deutsch_, Mar 31 2007