A185190 E.g.f. satisfies: A(x) = x + arcsinh(A(x))^2.

1, 2, 12, 112, 1440, 23648, 473088, 11164032, 303624960, 9351301632, 321717276672, 12228424826880, 508916576243712, 23016333612318720, 1124014843389984768, 58949533609403842560, 3304473379374295744512, 197167421810210663301120, 12476358616574849161101312

Offset: 1

Author

Paul D. Hanna, Feb 05 2012

Keywords

• nonn

Comments

Radius of convergence of the e.g.f. A(x) is r = 0.2767308231516982...,

where r and A(r) satisfy: r = A(r) - (1 + A(r)^2)/4 and

A(r) = sinh( sqrt(1 + A(r)^2)/2 ), so that A(r) = 0.6241087588791013...

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 112*x^4/4! + 1440*x^5/5! + ...
Related expansions:
arcsinh(A(x)) = x + 2*x^2/2! + 11*x^3/3! + 100*x^4/4! + 1269*x^5/5! + ...
arcsinh(A(x))^2 = 2*x^2/2! + 12*x^3/3! + 112*x^4/4! + 1440*x^5/5! + ...
Series expressions:
A(x) = x + arcsinh(x)^2 + d/dx arcsinh(x)^4/2! + d^2/dx^2 arcsinh(x)^6/3! + d^3/dx^3 arcsinh(x)^8/4! + ...
log(A(x)/x) = arcsinh(x)^2/x + d/dx (arcsinh(x)^4/x)/2! + d^2/dx^2 (arcsinh(x)^6/x)/3! + d^3/dx^3 (arcsinh(x)^8/x)/4! + ...
		

Crossrefs

Cf. A143136, A205886, A214431.

Programs

Mathematica

Rest[CoefficientList[InverseSeries[Series[x - ArcSinh[x]^2, {x, 0, 20}], x],x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 11 2014 *)
nmax = 20; A[] = 0; Do[A[x] = x + ArcSinh[A[x]]^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; Rest[CoefficientList[A[x], x] * Range[0, nmax]!] (* Vaclav Kotesovec, Aug 10 2025 *)

PARI

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

PARI

{a(n)=n!*polcoeff(serreverse(x-asinh(x+x*O(x^n))^2), n)}
for(n=1,26,print1(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, asinh(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, asinh(x+x*O(x^n))^(2*m)/x/m!)+x*O(x^n))); n!*polcoeff(A, n)}

Formula

E.g.f. satisfies:

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

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

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

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

a(n) ~ n^(n-1) * sqrt((1+s^2)/(2-s)) / (exp(n) * ((4*s-1-s^2)/4)^(n-1/2)), where s = 0.62410875887910134116443... is the root of the equation 2*log(s+sqrt(1+s^2)) = sqrt(1+s^2). - Vaclav Kotesovec, Jan 11 2014