A274113 - OEIS

A274113

Number of equivalence classes of ballot paths of length n for the string dud.

1, 1, 1, 1, 2, 3, 5, 7, 11, 16, 26, 39, 63, 95, 154, 235, 381, 585, 948, 1464, 2373, 3682, 5967, 9293, 15060, 23531, 38131, 59741, 96801, 152020, 246310, 387611, 627985, 990027, 1603893, 2532609, 4102726, 6487600, 10509114, 16639214, 26952186, 42722941, 69199472

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OFFSET

0,5

LINKS

Gheorghe Coserea, Table of n, a(n) for n = 0..300

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, proposition 3.6.

FORMULA

G.f. y satisfies: 0 = x^2*(1-x-x^2)*y^3 + 2*x*(1-3/2*x-x^2+x^3+x^4-x^5)*y^2 + (1-3*x-x^2+3*x^3+x^4-3*x^5)*y - 1+x^2-x^4. - Gheorghe Coserea, Jan 05 2017

MATHEMATICA

terms = 43; y[_] = 0; Do[y[x_] = (-1+x^2-x^4+(2x-3x^2-2x^3+2x^4+2x^5-2x^6) y[x]^2 + (x^2-x^3-x^4) y[x]^3)/(-1+3x+x^2-3x^3-x^4+ x^5) + O[x]^terms, terms]; CoefficientList[y[x], x] (* Jean-François Alcover, Oct 07 2018 *)

PROG

(PARI)

x='x; y='y;

Fxy = x^2*(1-x-x^2)*y^3 + 2*x*(1-3/2*x-x^2+x^3+x^4-x^5)*y^2 + (1-3*x-x^2+3*x^3+x^4-3*x^5)*y - 1+x^2-x^4;

seq(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);

Vec(y0);

};

seq(43) \\ Gheorghe Coserea, Jan 05 2017