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FORMULA
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a(n)=T(n,1), T(n,m)=1/3*(binomial(m,n-m)*9^(n-m)-sum(k=m+1..n-1, T(k,m)*sum(i=k..n, T(n,i)*T(i,k)))-sum(i=m+1..n-1, T(n,i)*T(i,m))), n>m, T(n,n)=1. - Vladimir Kruchinin, Mar 10 2012
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PROG
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(PARI) {a(n, m=3)=local(F=x+m*x^2+x*O(x^n), G); if(n<1, 0, for(k=3, n, G=F+x*O(x^k); for(i=1, m-1, G=subst(F, x, G)); F=F+((-polcoeff(G, k))/m)*x^k); return(polcoeff(F, n, x)))}
(Maxima)
T(n, m):=if n=m then 1 else 1/3*(binomial(m, n-m)*9^(n-m)-sum(T(k, m)*sum(T(n, i)*T(i, k), i, k, n), k, m+1, n-1)-sum(T(n, i)*T(i, m), i, m+1, n-1));
{T(n, k)=if(n==k, 1, if(n>k, 1/3*(binomial(k, n-k)*9^(n-k) - sum(j=k+1, n-1, T(j, k)*sum(i=j, n, T(n, i)*T(i, j)))-sum(i=k+1, n-1, T(n, i)*T(i, k)))))}
{a(n)=T(n, 1)} /* Efficiency can be improved if T(n, k) is stored in an array */
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