OFFSET
0,4
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
FORMULA
E.g.f. satisfies: A(x) = 1 + Integral [1 + x*A(x)^2] dx.
Let r be the radius of convergence of e.g.f. A(x), then: a(n)/n! ~ r^(n+2) where r=0.89757966985304971385345783421642045642527022484..., A(-r)=0.206876159989240..., A'(-r)=0.961585613659124...
r is the root of the equation 2*r^2*Hypergeometric0F1[1/3,-1/(9*r^3)] = Hypergeometric0F1[5/3,-1/(9*r^3)]. - Vaclav Kotesovec, Feb 23 2014
EXAMPLE
E.g.f.: A(x) = 1 + x + x^2/2! + 4*x^3/3! + 12*x^4/4! + 56*x^5/5! +...
A(x)^2 = 1 + 2*x + 4*x^2/2! + 14*x^3/3! + 62*x^4/4! + 312*x^5/5! +...
x*A(x)^2 = x + 4*x^2/2! + 12*x^3/3! + 56*x^4/4! + 310*x^5/5! +...
A'(x) = 1 + x + 4*x^2/2! + 12*x^3/3! + 56*x^4/4! + 310*x^5/5! +...
MATHEMATICA
CoefficientList[Series[-2*(Hypergeometric0F1[2/3, -x^3/9] + x*Hypergeometric0F1[4/3, -x^3/9]) / (-2*Hypergeometric0F1[1/3, -x^3/9] + x^2*Hypergeometric0F1[5/3, -x^3/9]), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Dec 21 2013 *)
PROG
(PARI) {a(n)=local(A=1+x); for(i=0, n, A=1+intformal(1+x*(A+x*O(x^n))^2)); n!*polcoeff(A, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 10 2008
STATUS
approved