OFFSET
0,3
COMMENTS
More generally, if A(x) = exp(p*x*exp(q*x*exp(r*x*A(x)))
where A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then
a(n,m) = Sum_{k=0..n} Sum_{j=0..k} p^(n-k)*q^(k-j)*r^j*C(n,k)*C(k,j)*m*(j+m)^(n-k-1)*(n-k)^(k-j)*(k-j)^j.
...
In general, if A(x) = F(x*G(x*H(x*A(x))) with F(0)=G(0)=H(0)=1,
where A(x)^m = Sum_{n>=0} a(n,m)*x^n, then
a(n,m) = Sum_{k=0..n} Sum_{j=0..k} {[x^(n-k)] F(x)^(j+m)*m/(j+m)} * {[x^(k-j)] G(x)^(n-k)} * {[x^j] H(x)^(k-j)}.
...
FORMULA
a(n) = Sum_{k=0..n} Sum_{j=0..k} 2^(k-j)*3^j*C(n,k)*C(k,j)*(j+1)^(n-k-1)*(n-k)^(k-j)*(k-j)^j.
EXAMPLE
E.g.f.: A(x) = 1 + x + 5*x^2/2! + 61*x^3/3! + 945*x^4/4! + 18401*x^5/5! +...
PROG
(PARI) {a(n, m=1, p=1, q=2, r=3)=n!*sum(k=0, n, sum(j=0, k, p^(n-k)*q^(k-j)*r^j*m*(j+m)^(n-k-1)/(n-k)!*(n-k)^(k-j)/(k-j)!*(k-j)^j/j!))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 27 2009
EXTENSIONS
Paul D. Hanna, Jun 28 2009
STATUS
approved