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%I #12 Feb 15 2014 02:31:27
%S 1,1,1,2,6,18,54,166,522,1670,5418,17786,58974,197226,664494,2253390,
%T 7685394,26345230,90721362,313682098,1088609142,3790610306,
%U 13239554790,46371693174,162835695258,573160873750,2021885799162,7146955776554
%N Number of Dyck paths of semilength n having no ascents of length 2 that start at an even level.
%C Column 0 of A114462.
%H Vincenzo Librandi, <a href="/A114464/b114464.txt">Table of n, a(n) for n = 0..200</a>
%F G.f.=[1-z+3z^2-z^3-(1-z)sqrt((1-4z+z^2)(1+z^2))]/(2z).
%F G.f. 1+x/(1-x)c(x^2/(1-x)^4), c(x) the g.f. of A000108; a(n+1)=sum{k=0..floor(n/2), C(n+2k,4k)C(k)}; - _Paul Barry_, May 31 2006
%F Conjecture: (n+1)*a(n) +(-5*n+3)*a(n-1) +2*(3*n-7)*a(n-2) +2*(-3*n+11)*a(n-3) +(5*n-27)*a(n-4) +(-n+7)*a(n-5)=0. - _R. J. Mathar_, Nov 26 2012
%F Recurrence: (n-3)*(n+1)*a(n) = (4*n^2 - 14*n + 9)*a(n-1) - (2*n^2 - 10*n + 15)*a(n-2) + (4*n^2 - 26*n + 39)*a(n-3) - (n-6)*(n-2)*a(n-4). - _Vaclav Kotesovec_, Feb 13 2014
%F a(n) ~ sqrt(2*sqrt(3)-3) * (2+sqrt(3))^n / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Feb 13 2014
%e a(4)=6 because we have UDUDUDUD, UDUUUDDD, UUUDDDUD, UUUDUDDD, UUUDDUDD and UUUUDDDD, where U=(1,1), D=(1,-1).
%p G:=(1-z+3*z^2-z^3-(1-z)*sqrt((1-4*z+z^2)*(1+z^2)))/2/z: Gser:=series(G,z=0,33): 1,seq(coeff(Gser,z^n),n=1..30);
%t CoefficientList[Series[(1-x+3*x^2-x^3-(1-x)*Sqrt[(1-4*x+x^2)*(1+x^2)])/2/x, {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 13 2014 *)
%Y Cf. A114462, A114463, A114465.
%K nonn
%O 0,4
%A _Emeric Deutsch_, Nov 29 2005
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