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%I #6 Jul 22 2022 11:44:52
%S 1,1,2,3,5,8,15,25,46,79,147,256,477,841,1570,2791,5217,9336,17467,
%T 31421,58830,106279,199103,360960,676545,1230185,2306642,4204931,
%U 7887045,14409480,27035135,49487641,92872062,170289575,319647235,586983680,1102027213,2026422689,3805138290
%N Number of length n left factors of Dyck paths having no UDUD's; here U=(1,1) and D=(1,-1).
%C a(n) = A191791(n,0).
%F G.f.: g(z)=C/(1-z*C), where C=C(z) is given by z^2*(1+z^2)*C^2-(1+z^2+z^4)*C+1+z^2=0.
%F Conjecture D-finite with recurrence (n+1)*a(n) -2*a(n-1) +2*(-n+1)*a(n-2) +4*(-1)*a(n-3) +5*(-n+3)*a(n-4) +4*a(n-5) +2*(-n+5)*a(n-6) +2*a(n-7) +(n-7)*a(n-8)=0. - _R. J. Mathar_, Jul 22 2022
%e a(4)=5 because we have UDUU, UUDD, UUDU, UUUD, and UUUU, where U=(1,1) and D=(1,-1) (the path UDUD does not qualify).
%p eq := z^2*(1+z^2)*C^2-(1+z^2+z^4)*C+1+z^2 = 0: C := RootOf(eq, C): g := C/(1-z*C): gser := series(g, z = 0, 42): seq(coeff(gser, z, n), n = 0 .. 38);
%Y Cf. A191791.
%K nonn
%O 0,3
%A _Emeric Deutsch_, Jun 18 2011
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