OFFSET
0,3
COMMENTS
Compare to: F(x) = exp(x + Integral Integral F(x) dx dx) holds when F(x) = 1/(1-sin(x)).
Compare to: G(x) = exp(x + Integral Integral G(x)^2 dx dx) holds when G(x) = 1/(1-x).
FORMULA
E.g.f.: 1/(2*cosh(sqrt(3)*x) - sqrt(3)*sinh(sqrt(3)*x) - 1)^(1/3). - Vaclav Kotesovec, Jan 05 2014
a(n) ~ n! * 2^(1/3) * 3^(n/2) / (GAMMA(2/3) * n^(1/3) * (log(2+sqrt(3)))^(n+2/3)). - Vaclav Kotesovec, Jan 05 2014
EXAMPLE
E.g.f.: A(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 34*x^4/4! + 209*x^5/5! +...
where
A(x)^3 = 1 + 3*x + 12*x^2/2! + 63*x^3/3! + 414*x^4/4! + 3267*x^5/5! +...
log(A(x)) = x + x^2/2! + 3*x^3/3! + 12*x^4/4! + 63*x^5/5! + 414*x^6/6! + 3267*x^7/7! +...
MATHEMATICA
CoefficientList[Series[(1/(-1 + 2*Cosh[Sqrt[3]*x] - Sqrt[3]*Sinh[Sqrt[3]*x]))^(1/3), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 05 2014 *)
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(x+intformal(intformal(A^3+x*O(x^n))))); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 21 2013
STATUS
approved