A274114 - OEIS

%I #27 Apr 25 2020 03:53:39

%S 1,1,1,2,4,8,17,37,81,180,405,917,2090,4795,11054,25589,59475,138712,

%T 324483,761163,1790028,4219139,9965328,23582735,55906518,132751359,

%U 315700152,751837207,1792853416,4280568845,10232005939,24484563844,58650123942,140625967460,337488663293,810641635789

%N Number of equivalence classes of Dyck paths of semilength n for the string uuu.

%H Gheorghe Coserea, <a href="/A274114/b274114.txt">Table of n, a(n) for n = 0..301</a>

%H K. Manes, A. Sapounakis, I. Tasoulas, P. Tsikouras, <a href="http://arxiv.org/abs/1510.01952">Equivalence classes of ballot paths modulo strings of length 2 and 3</a>, arXiv:1510.01952 [math.CO], 2015.

%F 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

%F a(n) ~ sqrt(51/4 + 577*sqrt(2)/64 + 19*sqrt(180250 + 127456*sqrt(2))/448) * (sqrt(13 + 16*sqrt(2))/2 - 1/2)^n / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Apr 25 2020

%t F[x_, y_] = x y^3 - (1 + 2x) y^2 + (1 + 3x) y - x;

%t 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];

%t seq[n_] := Module[{y = Y[n]}, ((1 + x y)/(1 - x (y-1)^2)) + O[x]^n // CoefficientList[#, x]&];

%t seq[36] (* _Jean-François Alcover_, Jul 27 2018, after _Gheorghe Coserea_ *)

%o (PARI)

%o x='x; y='y;

%o Fxy = x*y^3 - (1+2*x)*y^2 + (1+3*x)*y - x;

%o Y(N) = {

%o   my(y0 = 1 + O('x^N), y1=0);

%o   for (k = 1, N,

%o     y1 = y0 - subst(Fxy, y, y0)/subst(deriv(Fxy, y), y, y0);

%o     if (y1 == y0, break()); y0 = y1);

%o   y0;

%o };

%o seq(N) = my(y = Y(N)); Vec((1 + x*y)/(1 - x*(y-1)^2));

%o seq(35) \\ _Gheorghe Coserea_, Jan 05 2017

%Y Cf. A274110-A274115.

%K nonn,walk

%O 0,4

%A _N. J. A. Sloane_, Jun 17 2016

%E a(0)=1 prepended and more terms added by _Gheorghe Coserea_, Jan 05 2017