A274289 Number of equivalence classes of Dyck paths of semilength n for the string udu.

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

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-François Alcover, Jul 27 2018, after R. J. Mathar *)

Crossrefs

Cf. A274114, A274115.

Sequence in context: A238826 A048211 A098719 * A265023 A343291 A290996

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