A318007 - OEIS

%I #13 Oct 13 2020 05:18:21

%S 1,2,11,100,1321,22622,474851,11786920,337650001,10962556442,

%T 397804232891,15954966065740,700861670953081,33464274335643062,

%U 1725656338796874131,95578727117480612560,5658893822397686566561,356659432612090891074482,23841281202421071709150571,1684762749472172141605523380

%N E.g.f. A(x) satisfies: A(x) = sin(x) + cos(x)*A(x)^2 with A(0)=0.

%H Robert Israel, <a href="/A318007/b318007.txt">Table of n, a(n) for n = 1..368</a>

%F E.g.f. A(x) satisfies:

%F (1) A(x) = sin(x) + cos(x)*A(x)^2.

%F (2) A(x) = sin(x) * Sum_{n>=0} binomial(2*n,n)/(n+1) * sin(2*x)^n/2^n.

%F (3) A(x) = (1 - sqrt(1 - 2*sin(2*x))) / (2*cos(x)).

%F (4) A(x) = 2*sin(x) / (1 + sqrt(1 - 2*sin(2*x))).

%F 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

%e 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! + ...

%e such that A(x) = sin(x) + cos(x)*A(x)^2.

%p E:= (1 - sqrt(1 - 2*sin(2*x))) / (2*cos(x)):

%p S:= series(E,x,31):

%p seq(coeff(S,x,j)*j!,j=1..30); # _Robert Israel_, Aug 29 2018

%t 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 *)

%o (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)}

%o for(n=1, 25, print1(a(n), ", "))

%Y Cf. A318003, A318004, A318599.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Aug 28 2018