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%I #17 Oct 27 2021 09:27:13
%S 0,0,0,1,5,21,81,302,1107,4027,14608,52988,192501,701065,2560806,
%T 9384273,34504203,127288011,471102318,1749063906,6513268401,
%U 24323719461,91081800417,341929853235,1286711419527,4852902998951,18341683253676
%N Catalan number A000108(n) minus Motzkin number A001006(n).
%C Number of Dyck n-paths with at least one UUU. - _David Scambler_, Sep 17 2012
%F a(n) = A000108(n) - A001006(n).
%F Conjecture: -(n+1)*(n-3)*(n+2)^2*a(n) +3*(n+1)*(2*n^3-n^2-15*n+8)*a(n-1) -(n-1)*(5*n^3-41*n+48)*a(n-2) -6*(n-1)*(n-2)*(2*n-5)*(n+3)*a(n-3)=0, n>=6 - _R. J. Mathar_, Mar 04 2018
%p A001006 := proc(n) (3/2)^(n+2)*add( 3^(-k)*A000108(k-1)*binomial(k,n+2-k), k=1..n+2) ; end:
%p A153008 := proc(n) A000108(n)-A001006(n) ;
%p end:
%p seq(A153008(n),n=0..30) ; # _R. J. Mathar_, Jan 22 2009
%t MotzkinNumber = DifferenceRoot[Function[{y, n}, {(-3n-3)*y[n] + (-2n-5)*y[n+1] + (n+4)*y[n+2] == 0, y[0] == 1, y[1] == 1}]];
%t a[n_] := CatalanNumber[n] - MotzkinNumber[n];
%t Table[a[n], {n, 0, 26}] (* _Jean-François Alcover_, Oct 27 2021 *)
%Y Cf. A000108, A001006.
%K easy,nonn
%O 0,5
%A _Omar E. Pol_, Dec 20 2008
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