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A274289 Number of equivalence classes of Dyck paths of semilength n for the string udu. 1
1, 1, 2, 4, 9, 22, 54, 134, 335, 843, 2132, 5409, 13761, 35088, 89638, 229361, 587678, 1507586, 3871589, 9952087, 25604573, 65927447, 169875992, 438016016, 1130103976, 2917412699, 7535482753, 19473430909, 50347508572, 130228143004, 336985674038 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..30.

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.

MAPLE

G := 1 ;

T := 1 ;

for t from 1 to 40 do

    G := x*(1+G)+x^2*(1+x*G)*(1+x*(1+x*G))*G ;

    G := taylor(G, x=0, t+1) ;

    G := convert(G, polynom) ;

    T := (-x^2-x^3*T^3-x^2*T^2)/(x-1) ;

    T := taylor(T, x=0, t+1) ;

    T := convert(T, polynom) ;

    F := (x*(1-x)^2*(1+G+x*G)+x^5*(1+x*G)*G^2)/(1-x)/((1-x)^2+(x-2)*x^2*G)

               -x^4*(1-x+x^3)*(1+x*G)*G*T/(1-x)^2/(1-x+x^3-x*T) ;

    F := taylor(F, x=0, t+1) ;

    F := convert(F, polynom) ;

    for i from 0 to t do

        printf("%d, ", coeff(F, x, i)) ;

    od;

    print();

end do: # R. J. Mathar, Jun 21 2016

MATHEMATICA

G = 1; T = 1;

For[ t = 1 , t <= 40, t++,

G = x*(1 + G) + x^2*(1 + x*G)*(1 + x*(1 + x*G))*G + O[x]^(t+1) // Normal;

T = (-x^2 - x^3*T^3 - x^2*T^2)/(x - 1) + O[x]^(t+1) // Normal;

F = 1 + (x*(1 - x)^2*(1 + G + x*G) + x^5*(1 + x*G)*G^2)/(1 - x)/((1 - x)^2 + (x - 2)*x^2*G) - x^4*(1 - x + x^3)*(1 + x*G)*G*T/(1 - x)^2/(1 - x + x^3 - x*T) + O[x]^(t+1) // Normal;

];

CoefficientList[F, x] (* Jean-Fran├žois Alcover, Jul 27 2018, after R. J. Mathar *)

CROSSREFS

Cf. A274114, A274115.

Sequence in context: A238826 A048211 A098719 * A265023 A290996 A198520

Adjacent sequences:  A274286 A274287 A274288 * A274290 A274291 A274292

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jun 17 2016

EXTENSIONS

a(0)=1 prepended by Alois P. Heinz, Jul 27 2018

STATUS

approved

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Last modified November 19 22:34 EST 2019. Contains 329323 sequences. (Running on oeis4.)