OFFSET
0,4
FORMULA
From Paul D. Hanna, Mar 05 2009: (Start)
G.f.: A(x) = B(x) + sqrt(12*B(x) - 12 - 3*x^2)/3
where B(x) = (7-sqrt(1-12*x^2))/6 = A(x)*A(-x) = (A(x)+A(-x))/2 = 1 + x^2/(4-3*B(x)).
Lim_{n->infinity} a(2n)/a(2n-1) = 12^(1/3); lim_{n->infinity} a(2n+1)/a(2n) = 12^(2/3). (End)
EXAMPLE
G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 10*x^5 + 18*x^6 + 70*x^7 + ...
...
A(x) = 1 + x*exp( [A(x)-1] + [A(-x)-1]^2/2 + [A(x)-1]^3/3 + [A(-x)-1]^4/4 + ...).
PROG
(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1+x*exp(-sum(k=1, n, (subst(A, x, (-1)^k*x+x*O(x^n))-1)^k/k))); polcoeff(A, n)}
(PARI) {a(n)=local(B=(7-sqrt(1-12*x^2+x^2*O(x^n)))/6); polcoeff(B+sqrt(B^2-B), n)} \\ Paul D. Hanna, Mar 05 2009
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 04 2009
STATUS
approved