OFFSET
0,3
COMMENTS
The radius of convergence r of the e.g.f. A(x) satisfies: r = 1/(4*cos(r)) = limit (n+1)*a(n)/a(n+1) = 0.2585985822541... with A(r) = 1/(2*r) = 1.933498612565961...
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..365
FORMULA
E.g.f.: A(x) = (1 - sqrt(1 - 4*x*cos(x))) / (2*x).
a(n) = (-1)^floor(n/2)*(1 + (-1)^n)/2 + n*Sum_{k=0..n-1} binomial(n-1,k)*a(k)*a(n-1-k) for n >= 0.
a(n) ~ n! * sqrt(cos(r)-sin(r)*r)/(2*sqrt(Pi)*r^(n+1/2)*n^(3/2)), where r = 0.258598582254189... is defined in the comment. - Vaclav Kotesovec, Sep 21 2013
EXAMPLE
E.g.f.: A(x) = 1 + x + 3*x^2/2! + 24*x^3/3! + 265*x^4/4! + 3880*x^5/5! + ...
Related expansion:
A(x)^2 = 1 + 2*x + 8*x^2/2! + 66*x^3/3! + 776*x^4/4! + 11850*x^5/5! + ...
MATHEMATICA
CoefficientList[Series[(1-Sqrt[1-4*x*Cos[x]])/(2*x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 21 2013 *)
PROG
(PARI) {a(n)=n!*polcoeff((1 - sqrt(1 - 4*x*cos(x +O(x^(n+2))))) / (2*x), n)}
(PARI) {a(n)=(-1)^(n\2)*(1 + (-1)^n)/2 +n*sum(k=0, n-1, binomial(n-1, k)*a(k)*a(n-1-k))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 06 2011
STATUS
approved