A141118 G.f. A(x) satisfies: A(A(A(x))) = x + 9*x^2.

1, 3, -18, 189, -2430, 34020, -486972, 6786261, -86946372, 919825956, -5269375296, -80180038944, 3575424508272, -77211406919844, 1164244485947400, -12342809241883386, 102419678663170128, -2040575112980362980

Offset: 1

Author

Paul D. Hanna, Jun 05 2008

Keywords

• sign

Examples

			G.f.: A(x) = x + 3*x^2 - 18*x^3 + 189*x^4 - 2430*x^5 + 34020*x^6 -+ ...
A(A(x)) = x + 6*x^2 - 18*x^3 + 135*x^4 - 1296*x^5 + 13122*x^6 -+ ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A027436, A141119, A141120, A141121.

Programs

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));
makelist((T(n,1)),n,1,7); /* Vladimir Kruchinin, Mar 10 2012 */

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)))}

PARI

/* Using Vladimir Kruchinin's formula */
{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 */
for(n=1,20,print1(a(n),", ")) \\ Paul D. Hanna

Formula

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