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%I #9 May 04 2012 02:57:57
%S 1,1,3,12,60,346,2218,15444,115075,908155,7535185,65374018,590579575,
%T 5537249212,53742567000,538801229874,5570060420573,59288164937748,
%U 648934780013375,7295904025820975,84174136470742517,995682428049720830
%N G.f. satisfies: A(x) = Series_Reversion( x/(1 + A(x) + A(x)^2) ).
%F G.f. A(x) satisfies: A(x) = x*(1 + A(A(x)) + A(A(x))^2).
%F a(n)=T(n,1), T(n,j)=-sum(m=j..n-1, T(m,j)*sum(i=1..n-m, (sum(k=1..i, (-1)^k*binomial(k,i-k)*binomial(m+k-1,m-1)))*T(n-m,i))), n>j, T(n,n)=1. [_Vladimir Kruchinin_, May 04 2012]
%e G.f.: A(x) = x + x^2 + 3*x^3 + 12*x^4 + 60*x^5 + 346*x^6 + 2218*x^7 +...
%e A(A(x)) = x + 2*x^2 + 8*x^3 + 40*x^4 + 234*x^5 + 1526*x^6 +10816*x^7+...
%e A(A(x))^2 = x^2 + 4*x^3 + 20*x^4 + 112*x^5 + 692*x^6 + 4628*x^7 +...
%e x = A(x*[1 - A(x) + 2*A(x)^2 - 4*A(x)^3 + 9*A(x)^4 - 21*A(x)^5 +-...]).
%o (PARI) {a(n)=local(A=x); if(n<1, 0, for(i=1,n,A=serreverse(x/(1+A+A^2 +x*O(x^n)))); polcoeff(A, n))}
%o (Maxima) T(n,j):=if n=j then 1 else -sum(T(m,j)*sum((sum((-1)^k*binomial(k,i-k)*binomial(m+k-1,m-1),k,1,i))*T(n-m,i),i,1,n-m),m,j,n-1); makelist(T(n,1),n,1,10); [_Vladimir Kruchinin_, May 04 2012]
%Y Cf. A001006 (Motzkin numbers).
%K nonn
%O 1,3
%A _Paul D. Hanna_, May 15 2008
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