cp's OEIS Frontend

A274113 Number of equivalence classes of ballot paths of length n for the string dud.

%I A274113 #33 Oct 07 2018 08:06:15
%S A274113 1,1,1,1,2,3,5,7,11,16,26,39,63,95,154,235,381,585,948,1464,2373,3682,
%T A274113 5967,9293,15060,23531,38131,59741,96801,152020,246310,387611,627985,
%U A274113 990027,1603893,2532609,4102726,6487600,10509114,16639214,26952186,42722941,69199472
%N A274113 Number of equivalence classes of ballot paths of length n for the string dud.
%H A274113 Gheorghe Coserea, <a href="/A274113/b274113.txt">Table of n, a(n) for n = 0..300</a>
%H A274113 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, proposition 3.6.
%F A274113 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
%t A274113 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 *)
%o A274113 (PARI)
%o A274113 x='x; y='y;
%o A274113 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;
%o A274113 seq(N) = {
%o A274113   my(y0 = 1 + O('x^N), y1=0);
%o A274113   for (k = 1, N,
%o A274113     y1 = y0 - subst(Fxy, y, y0)/subst(deriv(Fxy, y), y, y0);
%o A274113     if (y1 == y0, break()); y0 = y1);
%o A274113   Vec(y0);
%o A274113 };
%o A274113 seq(43)  \\ _Gheorghe Coserea_, Jan 05 2017
%Y A274113 Cf. A274110, A274111, A274112, A274114, A274115.
%K A274113 nonn,walk
%O A274113 0,5
%A A274113 _N. J. A. Sloane_, Jun 17 2016
%E A274113 a(0)=1 prepended by _Gheorghe Coserea_, Jan 05 2017