OFFSET
1,2
LINKS
Robert Israel, Table of n, a(n) for n = 1..368
FORMULA
E.g.f. A(x) satisfies:
(1) A(x) = sin(x) + cos(x)*A(x)^2.
(2) A(x) = sin(x) * Sum_{n>=0} binomial(2*n,n)/(n+1) * sin(2*x)^n/2^n.
(3) A(x) = (1 - sqrt(1 - 2*sin(2*x))) / (2*cos(x)).
(4) A(x) = 2*sin(x) / (1 + sqrt(1 - 2*sin(2*x))).
a(n) ~ (sqrt(3) - 1) * 2^(2*n - 3/2) * 3^(n - 1/4) * n^(n-1) / (Pi^(n - 1/2) * exp(n)). - Vaclav Kotesovec, Oct 13 2020
EXAMPLE
E.g.f.: A(x) = x + 2*x^2/2! + 11*x^3/3! + 100*x^4/4! + 1321*x^5/5! + 22622*x^6/6! + 474851*x^7/7! + 11786920*x^8/8! + 337650001*x^9/9! + 10962556442*x^10/10! + ...
such that A(x) = sin(x) + cos(x)*A(x)^2.
MAPLE
E:= (1 - sqrt(1 - 2*sin(2*x))) / (2*cos(x)):
S:= series(E, x, 31):
seq(coeff(S, x, j)*j!, j=1..30); # Robert Israel, Aug 29 2018
MATHEMATICA
m = 21; A[x_] = (1 - Sqrt[1 - 2 Sin[2 x]] )/(2 Cos[x]); Rest[Range[0, m - 1]! * CoefficientList[A[x] + O[x]^m, x]] (* Jean-François Alcover, Apr 29 2019 *)
PROG
(PARI) {a(n) = my(A = 2*sin(x +x^2*O(x^n)) / (1 + sqrt(1 - 2*sin(2*x +x^2*O(x^n)))) ); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 28 2018
STATUS
approved