A214431 E.g.f. satisfies: A(x) = x + arcsin( A(x) )^2.

1, 2, 12, 128, 1920, 37088, 876288, 24478848, 789208320, 28841682432, 1178155579392, 53196745543680, 2630883967315968, 141432764970270720, 8211780541909352448, 512119065654535618560, 34141138126195739394048, 2422954554718763388764160, 182378340941780059828518912

Offset

1,2

Comments

Radius of convergence of A(x) is r = A(r) - (1-A(r)^2)/4 = 0.232465575..., where A(r) = 0.435130859... satisfies A(r) = sin(sqrt(1-A(r)^2)/2).

Links

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

Formula

E.g.f. satisfies:

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

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

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

(4) A'(x) = sqrt(1-A(x)^2) / (sqrt(1-A(x)^2) - 2*arcsin(A(x))).

a(n) ~ sqrt((1-s^2)/(2+s)) * n^(n-1) / (exp(n) * r^(n-1/2)), where r and s=A(r) were described above. - Vaclav Kotesovec, Jan 12 2014

Example

E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 128*x^4/4! + 1920*x^5/5! + ...
where
arcsin(A(x)) = x + 2*x^2/2! + 13*x^3/3! + 140*x^4/4! + 2109*x^5/5! + 40838*x^6/6! + 966393*x^7/7! + ...
arcsin(A(x))^2 = 2*x^2/2! + 12*x^3/3! + 128*x^4/4! + 1920*x^5/5! + 37088*x^6/6! + 876288*x^7/7! + ...
Related expansions:
A(x) = x + arcsin(x)^2 + d/dx arcsin(x)^4/2! + d^2/dx^2 arcsin(x)^6/3! + d^3/dx^3 arcsin(x)^8/4! + ...
log(A(x)/x) = arcsin(x)^2/x + d/dx (arcsin(x)^4/x)/2! + d^2/dx^2 (arcsin(x)^6/x)/3! + d^3/dx^3 (arcsin(x)^8/x)/4! + ...

Mathematica

Rest[CoefficientList[InverseSeries[Series[x - ArcSin[x]^2, {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 12 2014 *)

Prog

(PARI) {a(n)=n!*polcoeff(serreverse(x-asin(x+x*O(x^n))^2), n)}
(PARI) {a(n)=local(A=x); for(i=0, n, A=x + asin(A+x*O(x^n))^2); n!*polcoeff(A, 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, asin(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, asin(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. A143134, A185190.

Sequence in context: A003712 A143136 A214224 * A227461 A367374 A367853

Adjacent sequences: A214428 A214429 A214430 * A214432 A214433 A214434

Keyword

nonn

Author

Paul D. Hanna, Jul 17 2012

Status

approved