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A274111
Number of equivalence classes of ballot paths of length n for the string ddd.
3
1, 1, 1, 1, 1, 1, 2, 3, 5, 7, 10, 13, 20, 28, 45, 65, 101, 143, 222, 317, 500, 726, 1143, 1661, 2608, 3796, 5983, 8764, 13835, 20335, 32089, 47251, 74637, 110227, 174302, 258095, 408276, 605664, 958551, 1424659, 2256136, 3359446, 5322449, 7937666, 12580545
OFFSET
0,7
LINKS
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 3.2.
FORMULA
The g.f. satisfies x^2*(1-2*x+x^2-x^4)*A(x)^3 + 2*x*(1-x)^2*A(x)^2 + (1-3*x+x^2)*A(x) - 1 = 0. - R. J. Mathar, Jun 20 2016
MATHEMATICA
terms = 45; A[_]=0; Do[A[x_] = (1-2(-1+x)^2 x A[x]^2 + x^2 (-1+2x-x^2+x^4) A[x]^3)/(1-3x+x^2) + O[x]^terms, terms]; CoefficientList[A[x], x] (* Jean-François Alcover, Oct 07 2018 *)
PROG
(PARI)
x='x; y='y;
Fxy = x^2*(1-2*x+x^2-x^4)*y^3 + 2*x*(1-x)^2*y^2 + (1-3*x+x^2)*y - 1;
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(45) \\ Gheorghe Coserea, Jan 05 2017
CROSSREFS
KEYWORD
nonn,walk
AUTHOR
N. J. A. Sloane, Jun 17 2016
EXTENSIONS
a(0)=1 prepended by Gheorghe Coserea, Jan 05 2017
STATUS
approved