A143135 E.g.f. satisfies: A(x) = sin(x + A(x)^2) with A(0)=0.

1, 2, 11, 100, 1261, 20342, 399671, 9256840, 246907321, 7452534122, 251099460611, 9341422237420, 380293239870181, 16815919738248542, 802553031266952431, 41117164304824602640, 2250747364089063475441

Offset

1,2

Comments

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

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.: A(x) = sin(G(x)) where G(x) = x + A(x)^2 is the e.g.f. of A143134.

E.g.f. derivative: A'(x) = sqrt(1 - A(x)^2)/(1 - 2*A(x)*sqrt(1 - A(x)^2)).

a(n) ~ GAMMA(1/3) * 4^(n-1) * n^(n-5/6) / (3^(1/6) * sqrt(Pi) * exp(n) * (Pi-2)^(n-1/3)). - Vaclav Kotesovec, Jan 19 2014

Example

A(x) = x + 2*x^2/2! + 11*x^3/3! + 100*x^4/4! + 1261*x^5/5! +...
A(x) = sin(G(x)) where G(x) = x + A(x)^2 is the e.g.f. of A143134:
G(x) = x + 2*x^2/2! + 12*x^3/3! + 112*x^4/4! + 1440*x^5/5! +...

Mathematica

Rest[CoefficientList[InverseSeries[Series[-x^2 + ArcSin[x], {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 19 2014 *)

Prog

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

Crossrefs

Cf. A143134, A143137.

Sequence in context: A003579 A282640 A099169 * A205806 A220433 A318007

Adjacent sequences: A143132 A143133 A143134 * A143136 A143137 A143138

Keyword

nonn

Author

Paul D. Hanna, Jul 27 2008

Status

approved