A214222 - OEIS

%I #12 Jan 12 2014 11:16:10

%S 1,2,12,116,1560,26886,565376,14036392,401823360,13030976650,

%T 472154276352,18903994333212,828807273828352,39491616319733774,

%U 2032038033784995840,112293378446546611280,6632975513529162694656,417050432063319431036178,27809989478829060358799360

%N E.g.f. satisfies: A(x) = x/(1 - sin(A(x))).

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

%F (1) A(x - x*sin(x)) = x.

%F (2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n*sin(x)^n/n!.

%F (3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1)*sin(x)^n/n! ).

%F a(n) = n*A201627(n-1).

%F a(n) = (n-1)! * [x^n] x/(1 - sin(x))^n.

%F a(n) ~ sqrt((1-t)/(2+t)) * n^(n-1) * (sqrt(1-t^2)/(1-t)^2)^n / exp(n), where t = 0.527766122670442778... is the root of the equation t = sin(sqrt((1-t)/(1+t))). - _Vaclav Kotesovec_, Jan 12 2014

%e E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 116*x^4/4! + 1560*x^5/5! +...

%e Related expansions:

%e A(x) = x + x*sin(x) + d/dx x^2*sin(x)^2/2! + d^2/dx^2 x^3*sin(x)^3/3! + d^3/dx^3 x^4*sin(x)^4/4! +...

%e log(A(x)/x) = sin(x) + d/dx x*sin(x)^2/2! + d^2/dx^2 x^2*sin(x)^3/3! + d^3/dx^3 x^3*sin(x)^4/4! +...

%e A(x)/x = 1 + x + 4*x^2/2! + 29*x^3/3! + 312*x^4/4! + 4481*x^5/5! + 80768*x^6/6! +...+ A201627(n)*x^n/n! +...

%e sin(A(x)) = x + 2*x^2/2! + 11*x^3/3! + 104*x^4/4! + 1381*x^5/5! + 23616*x^6/6! + 493975*x^7/7! + 12216448*x^8/8! +...

%t Rest[CoefficientList[InverseSeries[Series[x - x*Sin[x],{x,0,20}],x],x] * Range[0,20]!] (* _Vaclav Kotesovec_, Jan 12 2014 *)

%o (PARI) {a(n)=(n-1)!*polcoeff(x/(1 - sin(x+x*O(x^n)))^n,n)}

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

%o (PARI) {a(n)=n!*polcoeff(serreverse(x-x*sin(x+x*O(x^n))), n)}

%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}

%o {a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, x^m*sin(x+x*O(x^n))^m/m!)); n!*polcoeff(A, n)}

%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}

%o {a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, x^(m-1)*sin(x+x*O(x^n))^m/m!)+x*O(x^n))); n!*polcoeff(A, n)}

%Y Cf. A201627, A214223, A214224, A214225.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jul 07 2012