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%I #10 Jul 22 2022 12:51:19
%S 1,1,2,3,6,12,25,53,114,249,550,1227,2760,6253,14256,32682,75293,
%T 174224,404741,943622,2207135,5177817,12179904,28722736,67890481,
%U 160812128,381671061,907529504,2161622683,5157014539,12321750366,29482362166
%N Number of Dyck paths with no UUU's and no DDD's of semilength n and having no UUDUDD's (U=(1,1), D=(1,-1)).
%C a(n) = A162984(n,0).
%H Paul Barry, <a href="https://arxiv.org/abs/1910.00875">Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials</a>, arXiv:1910.00875 [math.CO], 2019.
%F G.f. = G(z) satisfies G = 1 + zG + z^2*G + z^3*G(G-1).
%F D-finite with recurrence (n+3)*a(n) +(-2*n-3)*a(n-1) -n*a(n-2) +(-n+3)*a(n-4) +(-2*n+9)*a(n-5) +(n-6)*a(n-6)=0. - _R. J. Mathar_, Jul 22 2022
%e a(3)=3 because we have UDUDUD, UDUUDD, and UUDDUD.
%p G := ((1-z-z^2+z^3-sqrt(1-2*z-z^2-z^4-2*z^5+z^6))*1/2)/z^3: Gser := series(G, z = 0, 36): seq(coeff(Gser, z, n), n = 0 .. 31);
%Y Cf. A162984.
%K nonn
%O 0,3
%A _Emeric Deutsch_, Oct 11 2009