OFFSET
1,2
COMMENTS
Given e.g.f. A(x), then A(Pi/6) = Pi/2, where Pi/6 is the radius of convergence.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..250
FORMULA
E.g.f. A(x) satisfies: A(-A(-x)) = x.
The n-th iteration of e.g.f. A(x) equals: arcsin(x) o x/(1-n*x) o sin(x) = arcsin( sin(x)/(1-n*sin(x)) ).
a(n) ~ 2^n * 3^(n-1/4) * n^(n-1) / (Pi^(n-1/2) * exp(n)). - Vaclav Kotesovec, Apr 05 2016
EXAMPLE
E.g.f.: A(x) = x + 2*x^2/2! + 6*x^3/3! + 28*x^4/4! + 180*x^5/5! +...
where the initial iterations of e.g.f. A(x) begin:
A(A(x)) = arcsin( sin(x)/(1-2*sin(x)) ); more explicitly,
A(A(x)) = x + 4*x^2/2! + 24*x^3/3! + 200*x^4/4! + 2160*x^5/5! +...
A(A(A(x))) = arcsin( sin(x)/(1-3*sin(x)) ); more explicitly,
A(A(A(x))) = x + 6*x^2/2! + 54*x^3/3! + 660*x^4/4! + 10260*x^5/5! +...
A(A(A(A(x)))) = arcsin( sin(x)/(1-4*sin(x)) ); more explicitly,
A(A(A(A(x)))) = x + 8*x^2/2! + 96*x^3/3! + 1552*x^4/4! + 31680*x^5/5! +...
PROG
(PARI) {a(n)=n!*polcoeff(subst(asin(x+x*O(x^n)), x, subst(x/(1-x), x, sin(x+x*O(x^n)))), n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 29 2011
STATUS
approved