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%I #18 Jul 27 2018 12:36:07
%S 1,1,2,4,9,22,54,134,335,843,2132,5409,13761,35088,89638,229361,
%T 587678,1507586,3871589,9952087,25604573,65927447,169875992,438016016,
%U 1130103976,2917412699,7535482753,19473430909,50347508572,130228143004,336985674038
%N Number of equivalence classes of Dyck paths of semilength n for the string udu.
%H 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.
%p G := 1 ;
%p T := 1 ;
%p for t from 1 to 40 do
%p G := x*(1+G)+x^2*(1+x*G)*(1+x*(1+x*G))*G ;
%p G := taylor(G,x=0,t+1) ;
%p G := convert(G,polynom) ;
%p T := (-x^2-x^3*T^3-x^2*T^2)/(x-1) ;
%p T := taylor(T,x=0,t+1) ;
%p T := convert(T,polynom) ;
%p 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)
%p -x^4*(1-x+x^3)*(1+x*G)*G*T/(1-x)^2/(1-x+x^3-x*T) ;
%p F := taylor(F,x=0,t+1) ;
%p F := convert(F,polynom) ;
%p for i from 0 to t do
%p printf("%d,",coeff(F,x,i)) ;
%p od;
%p print();
%p end do: # _R. J. Mathar_, Jun 21 2016
%t G = 1; T = 1;
%t For[ t = 1 , t <= 40, t++,
%t G = x*(1 + G) + x^2*(1 + x*G)*(1 + x*(1 + x*G))*G + O[x]^(t+1) // Normal;
%t T = (-x^2 - x^3*T^3 - x^2*T^2)/(x - 1) + O[x]^(t+1) // Normal;
%t 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;
%t ];
%t CoefficientList[F, x] (* _Jean-François Alcover_, Jul 27 2018, after _R. J. Mathar_ *)
%Y Cf. A274114, A274115.
%K nonn
%O 0,3
%A _N. J. A. Sloane_, Jun 17 2016
%E a(0)=1 prepended by _Alois P. Heinz_, Jul 27 2018
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