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%I #11 Jul 26 2022 14:14:41
%S 1,1,2,4,10,27,79,240,750,2387,7711,25214,83315,277799,933596,3159187,
%T 10755190,36811479,126594819,437220744,1515844359,5273760446,
%U 18406122609,64426136558,226108087891,795486834627,2804993559426
%N Number of Dyck paths of semilength n having no ascents of length 3.
%C Also number of ordered trees with n edges that have no vertices of outdegree 3.
%F G.f. G satisfies z^4*G^4-z^3*G^3+zG^2-G+1=0.
%F a(n) = 1/n*sum(j=ceiling((n+1)/2)..n, C(n,j)*C(4*j-2*n-2,j-1)*(-1)^(n-j)) n>0. [From _Vladimir Kruchinin_, Mar 07 2011]
%F D-finite with recurrence 2*n*(26405927*n-73197215)*(2*n+3)*(n+1)*a(n) +2*n*(2*n+1)*(26405927*n^2-273126414*n+480676927)*a(n-1) +4*(-793701648*n^4+4928830819*n^3-11073984912*n^2+10499531162*n-3092762541)*a(n-2) +2*(375778330*n^4-3447814000*n^3+22123257551*n^2-60324066977*n+51211836006)*a(n-3) +2*(12664700570*n^4-145150764350*n^3+621947195977*n^2-1179232268341*n+833841845214)*a(n-4) -3*(n-4)*(11017381441*n^3-111829680906*n^2+390445674963*n-461862831838)*a(n-5) -(n-4)*(n-5)*(30888861033*n^2-148676625095*n+156786419682)*a(n-6) +3206*(n-5)*(n-6)*(18970222*n-55906401)*(n-4)*a(n-7)=0. - _R. J. Mathar_, Jul 26 2022
%e a(3) = 4 because we have UDUDUD, UDUUDD, UUDDUD and UUDUDD, where U=(1,1), D=(1,-1).
%p Order:=36: Y:=solve(series((Y-Y^2)/(1-Y^3+Y^4),Y)=z,Y): seq(coeff(Y,z^n),n=1..32); #(Y=zG)
%o (Maxima) a114507(n):= 1/n*sum(binomial(n,j)*binomial(4*j-2*n-2, j-1) *(-1)^(n-j),j,ceiling((n+1)/2),n); [_Vladimir Kruchinin_, Mar 07 2011].
%Y Cf. A102403, A114506, A114509.
%K nonn
%O 0,3
%A _Emeric Deutsch_, Dec 03 2005