A143134 E.g.f. satisfies: A(x) = x + sin( A(x) )^2 with A(0)=0.

1, 2, 12, 112, 1440, 23552, 467712, 10926592, 293544960, 8914583552, 301957742592, 11285975498752, 461367611228160, 20477098870833152, 980591931131953152, 50393637174029320192, 2766350676943951626240

Offset

1,2

Comments

Radius of convergence of A(x) is r = Pi/4 - 1/2, with A(r) = Pi/4.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..370

V. Kotesovec, Asymptotic of implicit functions if Fww = 0, Jan 19 2014

Formula

E.g.f. satisfies:

(1) A(x) = Series_Reversion( x - sin(x)^2 ).

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

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

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

E.g.f. derivative: A'(x) = 1/(1 - 2*sqrt(A(x)-x)*sqrt(1+x-A(x))); thus A'(x) = 1/(1 - sin(2*A(x))).

Let f(x) = 1/(1-sin(2*x)). Then a(n) = (f(x)*d/dx)^(n-1) f(x) evaluated at x = 0. - Peter Bala, Oct 12 2011

a(n) ~ GAMMA(1/3) * 2^(2*n-3/2) * n^(n-5/6) / (3^(1/6) * sqrt(Pi) * exp(n) * (Pi-2)^(n-1/3)). - Vaclav Kotesovec, Jan 18 2014

Example

A(x) = x + 2*x^2/2! + 12*x^3/3! + 112*x^4/4! + 1440*x^5/5! +...
sin(A(x)) = G(x) is the e.g.f. of A143135:
G(x) = x + 2*x^2/2! + 11*x^3/3! + 100*x^4/4! + 1261*x^5/5! +...
G(x)^2 = 2*x^2/2! + 12*x^3/3! + 112*x^4/4! + 1440*x^5/5! +...
Related expansions:
A(x) = x + sin(x)^2 + d/dx sin(x)^4/2! + d^2/dx^2 sin(x)^6/3! + d^3/dx^3 sin(x)^8/4! +...
log(A(x)/x) = sin(x)^2/x + d/dx (sin(x)^4/x)/2! + d^2/dx^2 (sin(x)^6/x)/3! + d^3/dx^3 (sin(x)^8/x)/4! +...

Mathematica

Rest[CoefficientList[InverseSeries[Series[x - Sin[x]^2, {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 18 2014 *)

Prog

(PARI) {a(n)=local(A=x); for(i=0, n, A=x + sin(A)^2); n!*polcoeff(A, n)}
(PARI) {a(n)=n!*polcoeff(serreverse(x-sin(x+x*O(x^n))^2), 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, sin(x+x*O(x^n))^(2*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, sin(x+x*O(x^n))^(2*m)/x/m!)+x*O(x^n))); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))

Crossrefs

Cf. A143135, A143136, A214770.

Sequence in context: A292187 A124213 A365558 * A214225 A185190 A227460

Adjacent sequences: A143131 A143132 A143133 * A143135 A143136 A143137

Keyword

nonn

Author

Paul D. Hanna, Jul 27 2008

Status

approved