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 A274114 Number of equivalence classes of Dyck paths of semilength n for the string uuu. 5
 1, 1, 1, 2, 4, 8, 17, 37, 81, 180, 405, 917, 2090, 4795, 11054, 25589, 59475, 138712, 324483, 761163, 1790028, 4219139, 9965328, 23582735, 55906518, 132751359, 315700152, 751837207, 1792853416, 4280568845, 10232005939, 24484563844, 58650123942, 140625967460, 337488663293, 810641635789 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS Gheorghe Coserea, Table of n, a(n) for n = 0..301 K. Manes, A. Sapounakis, I. Tasoulas, P. Tsikouras, Equivalence classes of ballot paths modulo strings of length 2 and 3, arXiv:1510.01952 [math.CO], 2015. FORMULA A(x) = (1 + x*y)/(1 - x*(y-1)^2), where 0 = x*y^3 - (1+2*x)*y^2 + (1+3*x)*y - x with y(0)=1. - Gheorghe Coserea, Jan 05 2017 MATHEMATICA F[x_, y_] = x y^3 - (1 + 2x) y^2 + (1 + 3x) y - x; Y[n_] := Module[{y0 = 1, y1 = 0}, For[k = 1, k <= n, k++, y1 = y0 - F[x, y0] / (D[F[x, y], y] /. y -> y0) + O[x]^n // Normal; If[y1 == y0, Break[]]; y0 = y1]; y0]; seq[n_] := Module[{y = Y[n]}, ((1 + x y)/(1 - x (y-1)^2)) + O[x]^n // CoefficientList[#, x]&]; seq[36] (* Jean-François Alcover, Jul 27 2018, after Gheorghe Coserea *) PROG (PARI) x='x; y='y; Fxy = x*y^3 - (1+2*x)*y^2 + (1+3*x)*y - x; Y(N) = {   my(y0 = 1 + O('x^N), y1=0);   for (k = 1, N,     y1 = y0 - subst(Fxy, y, y0)/subst(deriv(Fxy, y), y, y0);     if (y1 == y0, break()); y0 = y1);   y0; }; seq(N) = my(y = Y(N)); Vec((1 + x*y)/(1 - x*(y-1)^2)); seq(35) \\ Gheorghe Coserea, Jan 05 2017 CROSSREFS Cf. A274110-A274115. Sequence in context: A129987 A132275 A136671 * A024557 A199409 A025241 Adjacent sequences:  A274111 A274112 A274113 * A274115 A274116 A274117 KEYWORD nonn,walk AUTHOR N. J. A. Sloane, Jun 17 2016 EXTENSIONS a(0)=1 prepended and more terms added by Gheorghe Coserea, Jan 05 2017 STATUS approved

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Last modified December 6 06:34 EST 2019. Contains 329784 sequences. (Running on oeis4.)