OFFSET
0,2
FORMULA
E.g.f. A(x) satisfies:
(1) A'(x) = -1 + A(x) + 2*A(x)^2.
(2) A(x) = exp(x + Integral 2*A(x) - 1/A(x) dx).
(3) A(x) = 1 + Series_Reversion( Integral 1/((2+x)*(1+2*x)) dx ).
a(n) ~ n!/2 * (3/(2*log(2)))^(n+1). - Vaclav Kotesovec, Dec 19 2013
EXAMPLE
E.g.f.: A(x) = 1 + 2*x + 10*x^2/2! + 66*x^3/3! + 570*x^4/4! + 6162*x^5/5! +...
MATHEMATICA
CoefficientList[Series[(2+E^(3*x))/(4-E^(3*x)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Dec 19 2013 *)
PROG
(PARI) {a(n)=local(A=1+x, X=x+x*O(x^n)); n!*polcoeff((2+exp(3*X))/(4-exp(3*X)), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=exp(x+intformal(2*A-1/A+x*O(x^n)))); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 18 2013
STATUS
approved