OFFSET
1,3
FORMULA
G.f. A(x) satisfies: A(x) = x*(1 + A(A(x)) + A(A(x))^2).
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]
EXAMPLE
G.f.: A(x) = x + x^2 + 3*x^3 + 12*x^4 + 60*x^5 + 346*x^6 + 2218*x^7 +...
A(A(x)) = x + 2*x^2 + 8*x^3 + 40*x^4 + 234*x^5 + 1526*x^6 +10816*x^7+...
A(A(x))^2 = x^2 + 4*x^3 + 20*x^4 + 112*x^5 + 692*x^6 + 4628*x^7 +...
x = A(x*[1 - A(x) + 2*A(x)^2 - 4*A(x)^3 + 9*A(x)^4 - 21*A(x)^5 +-...]).
PROG
(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))}
(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]
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 15 2008
STATUS
approved