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%I #9 Jul 22 2022 12:52:50
%S 1,1,2,3,5,9,17,34,70,147,313,673,1459,3185,6995,15445,34265,76342,
%T 170744,383214,862814,1948299,4411167,10011973,22775773,51920833,
%U 118593423,271376295,622047011,1428128025,3283679333,7560750299
%N Number of Dyck paths with no UUU's and no DDD's, of semilength n having no peak plateaux (U=(1,1), D=(1,-1)).
%C A peak plateau is a run of consecutive peaks that is preceded by an upstep U and followed by a down step D; a peak consists of an upstep followed by a downstep.
%F a(n) = A166285(n,0).
%F G.f.: G(z) satisfies G = 1 + zG + z^2*G + z^3*G[G - 1/(1-z)].
%F D-finite with recurrence (n+3)*a(n) +(-5*n-9)*a(n-1) +2*(4*n+3)*a(n-2) -4*n*a(n-3) +2*(-2*n+9)*a(n-5) +2*(4*n-21)*a(n-6) +4*(-n+6)*a(n-7)=0. - _R. J. Mathar_, Jul 22 2022
%e a(3)=3 because we have UDUDUD, UDUUDD, and UUDDUD (UUDUDD is a peak plateau).
%p F := RootOf(G = 1+z*G+z^2*G+z^3*G*(G-1/(1-z)), G): Fser := series(F, z = 0, 35): seq(coeff(Fser, z, n), n = 0 .. 32);
%Y Cf. A166285.
%K nonn
%O 0,3
%A _Emeric Deutsch_, Oct 12 2009
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