A143136 E.g.f. satisfies: A(x) = x + sinh( A(x) )^2.

1, 2, 12, 128, 1920, 36992, 870912, 24232448, 777999360, 28309164032, 1151292628992, 51750540443648, 2547747292446720, 136336755956252672, 7879446478581399552, 489119124160488931328, 32456290094449950720000

Offset

1,2

Comments

Radius of convergence is r = log(sqrt(2)+1)/2 - (sqrt(2)-1)/2 = 0.2335800...,

where A(r) = log(1+sqrt(2))/2 = arcsinh(1)/2 = 0.44068679...

Links

Alois P. Heinz, Table of n, a(n) for n = 1..150

Formula

E.g.f.: A(x) = Series_Reversion( x - sinh(x)^2 ).

E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) sinh(x)^(2*n)/n!.

E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (sinh(x)^(2*n)/x)/n! ).

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

a(n) ~ 2^(n-5/4) * n^(n-1) / (exp(n) * (1-sqrt(2)+log(1+sqrt(2)))^(n-1/2)). - Vaclav Kotesovec, Jan 08 2014

Example

A(x) = x + 2*x^2/2! + 12*x^3/3! + 128*x^4/4! + 1920*x^5/5! + ...
sinh(A(x)) = G(x) is the e.g.f. of A143137:
G(x) = x + 2*x^2/2! + 13*x^3/3! + 140*x^4/4! + 2101*x^5/5! + ...
Related expansions:
A(x) = x + sinh(x)^2 + d/dx sinh(x)^4/2! + d^2/dx^2 sinh(x)^6/3! + d^3/dx^3 sinh(x)^8/4! + ...
log(A(x)/x) = sinh(x)^2/x + d/dx (sinh(x)^4/x)/2! + d^2/dx^2 (sinh(x)^6/x)/3! + d^3/dx^3 (sinh(x)^8/x)/4! +...

Mathematica

Rest[CoefficientList[InverseSeries[Series[x - Sinh[x]^2, {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 08 2014 *)

Prog

(PARI) {a(n)=n!*polcoeff(serreverse(x-sinh(x+x*O(x^n))^2), n)}
(PARI) {a(n)=local(A=x); for(i=0, n, A=x + sinh(A)^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, sinh(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, sinh(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, A143137, A213643.

Sequence in context: A349268 A366821 A003712 * A214224 A214431 A227461

Adjacent sequences: A143133 A143134 A143135 * A143137 A143138 A143139

Keyword

nonn

Author

Paul D. Hanna, Jul 27 2008

Status

approved