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

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

Crossrefs

Cf. A274110, A274111, A274112, A274114, A274115.

Sequence in context: A130137 A218022 A091980 * A005685 A141656 A092180

Adjacent sequences: A274110 A274111 A274112 * A274114 A274115 A274116

Keyword

nonn,walk

Author

N. J. A. Sloane, Jun 17 2016

Extensions

a(0)=1 prepended by Gheorghe Coserea, Jan 05 2017

Status

approved