A214222 E.g.f. satisfies: A(x) = x/(1 - sin(A(x))).

1, 2, 12, 116, 1560, 26886, 565376, 14036392, 401823360, 13030976650, 472154276352, 18903994333212, 828807273828352, 39491616319733774, 2032038033784995840, 112293378446546611280, 6632975513529162694656, 417050432063319431036178, 27809989478829060358799360

Offset

1,2

Links

Table of n, a(n) for n=1..19.

Formula

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

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

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

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

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

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

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

Example

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

Mathematica

Rest[CoefficientList[InverseSeries[Series[x - x*Sin[x], {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 12 2014 *)

Prog

(PARI) {a(n)=(n-1)!*polcoeff(x/(1 - sin(x+x*O(x^n)))^n, n)}
for(n=1, 25, print1(a(n), ", "))
(PARI) {a(n)=n!*polcoeff(serreverse(x-x*sin(x+x*O(x^n))), n)}
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{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)}
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{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)}

Crossrefs

Cf. A201627, A214223, A214224, A214225.

Sequence in context: A225797 A302286 A035051 * A227459 A377716 A372200

Adjacent sequences: A214219 A214220 A214221 * A214223 A214224 A214225

Keyword

nonn

Author

Paul D. Hanna, Jul 07 2012

Status

approved