This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A191792 #6 Jul 22 2022 11:44:52 %S A191792 1,1,2,3,5,8,15,25,46,79,147,256,477,841,1570,2791,5217,9336,17467, %T A191792 31421,58830,106279,199103,360960,676545,1230185,2306642,4204931, %U A191792 7887045,14409480,27035135,49487641,92872062,170289575,319647235,586983680,1102027213,2026422689,3805138290 %N A191792 Number of length n left factors of Dyck paths having no UDUD's; here U=(1,1) and D=(1,-1). %C A191792 a(n) = A191791(n,0). %F A191792 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 A191792 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 A191792 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 A191792 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 A191792 Cf. A191791. %K A191792 nonn %O A191792 0,3 %A A191792 _Emeric Deutsch_, Jun 18 2011