A191792 Number of length n left factors of Dyck paths having no UDUD's; here U=(1,1) and D=(1,-1).
1, 1, 2, 3, 5, 8, 15, 25, 46, 79, 147, 256, 477, 841, 1570, 2791, 5217, 9336, 17467, 31421, 58830, 106279, 199103, 360960, 676545, 1230185, 2306642, 4204931, 7887045, 14409480, 27035135, 49487641, 92872062, 170289575, 319647235, 586983680, 1102027213, 2026422689, 3805138290
Offset: 0
Keywords
Examples
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).
Crossrefs
Cf. A191791.
Programs
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Maple
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);
Formula
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.
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
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