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
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
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
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