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%I #17 Jul 23 2017 12:17:49
%S 0,0,2,9,41,189,880,4131,19522,92763,442798,2121795,10200477,49176639,
%T 237661176,1151032005,5585185425,27146751885,132145210270,
%U 644128990155,3143590707235,15358979381175,75117256339240,367723284610905
%N Number of ascents of length at least 2 in all skew Dyck paths of semilength n.
%C 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.
%H Vincenzo Librandi, <a href="/A128752/b128752.txt">Table of n, a(n) for n = 0..300</a>
%H 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 a(n) = Sum_{k>=0} k*A128751(n,k).
%F G.f.: (1/2)(1-2z)sqrt((1-z)/(1-5z)) - 1/2.
%F Recurrence: n*(3*n-1)*a(n) = 18*(n-1)*n*a(n-1) - 5*(n-3)*(3*n+2)*a(n-2). - _Vaclav Kotesovec_, Nov 20 2012
%F a(n) ~ 3*5^(n-3/2)/sqrt(Pi*n). - _Vaclav Kotesovec_, Nov 20 2012
%e a(2)=2 because we have UUDD and UUDL.
%p G:=(1/2)*(1-2*z)*sqrt((1-z)/(1-5*z))-1/2: Gser:=series(G,z=0,30): seq(coeff(Gser,z,n),n=0..27);
%t CoefficientList[Series[1/2*(1-2*x)*Sqrt[(1-x)/(1-5*x)]-1/2, {x, 0, 20}], x] (* _Vaclav Kotesovec_, Nov 20 2012 *)
%Y Cf. A128751.
%K nonn
%O 0,3
%A _Emeric Deutsch_, Mar 31 2007
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