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%I #4 Apr 30 2014 01:32:50
%S 0,1,1,1,2,4,9,21,50,122,303,764,1950,5028,13077,34265,90368,239696,
%T 639011,1711281,4601504,12418610,33627437,91334429,248761858,
%U 679271970,1859207099,5099872111,14017476257,38600963255,106485177331
%N Expansion of solution to xA(x)(A(x)-x+1)=A(xA(x)).
%C Series reversion of g.f. A(x) is -A(-x).
%C For n>=1, a(n) = number of Dyck (n-1)-paths with no matching UU-DD pairs and no DDDDs. Also, a(n) = number of noncrossing partitions of [n-1] in which no block contains two consecutive integers and all blocks have size <=3. For example, a(5)=4 counts 14-2-3, 1-24-3, 1-2-3-4, 13-2-4, where dashes separate the blocks. - _David Callan_, Jul 27 2007
%F G.f. A(x) satisfies A(x)= x +(A(x*A(x))-x*A(x))/(x*A(x)).
%o (PARI) a(n)=local(A=x+x^2+O(x^3));for(i=3,n,A=x+subst((A-x)/x,x,x*A));polcoeff(A,n)
%K nonn
%O 0,5
%A _Michael Somos_, Jan 23 2004
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