OFFSET
0,3
FORMULA
E.g.f. A(x) satisfies:
(1a) A(x) = cos(x) + sin(x)*A(x)^2.
(1b) A(x) = cos(x) * Sum_{n>=0} binomial(2*n,n)/(n+1) * sin(2*x)^n/2^n.
(2a) A(x) = (1 - sqrt(1 - 2*sin(2*x))) / (2*sin(x)).
(2b) A(x) = 2*cos(x) / (1 + sqrt(1 - 2*sin(2*x))).
a(n) ~ (1 + sqrt(3)) * 2^(2*n - 3/2) * 3^(n - 1/4) * n^(n-1) / (exp(n) * Pi^(n - 1/2)). - Vaclav Kotesovec, Aug 21 2018
EXAMPLE
E.g.f.: A(x) = 1 + x + 3*x^2/2! + 23*x^3/3! + 249*x^4/4! + 3601*x^5/5! + 65163*x^6/6! + 1420103*x^7/7! + 36240369*x^8/8! + 1060638241*x^9/9! + ...
such that
A(x) = cos(x) + sin(x)*A(x)^2.
RELATED SERIES.
log(A(x)) = x + 2*x^2/2! + 16*x^3/3! + 160*x^4/4! + 2240*x^5/5! + 39392*x^6/6! + 841216*x^7/7! + 21130240*x^8/8! + 610734080*x^9/9! + ...
A(x)^2 = 1 + 2*x + 8*x^2/2! + 64*x^3/3! + 736*x^4/4! + 11072*x^5/5! + 206528*x^6/6! + 4607104*x^7/7! + 119766016*x^8/8! + ...
PROG
(PARI) {a(n) = my(A = 2*cos(x +x^2*O(x^n)) / (1 + sqrt(1 - 2*sin(2*x +x^2*O(x^n)))) ); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 21 2018
STATUS
approved