A226760 E.g.f.: A(x) = x + tanh(A(x)^2).

1, 2, 12, 120, 1680, 30000, 651840, 16692480, 492307200, 16432899840, 612414190080, 25205181281280, 1135434972672000, 55567263749437440, 2935699697464934400, 166525788640142131200, 10094443872515806003200, 651211844046521735577600, 44547029929095149430374400

Offset

1,2

Links

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

Formula

E.g.f.: Series_Reversion(x - tanh(x^2)).

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

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

a(n) ~ n^(n-1) * sqrt(r*s/(1 - 4*s^2*(s-r))) / (exp(n) * r^n), where s = 0.5456469378931069437... is the root of the equation 2*s = cosh(s^2)^2, and r = s - tanh(s^2) = 0.2564125251556591672... - Vaclav Kotesovec, Jan 23 2014

Example

E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 120*x^4/4! + 1680*x^5/4! +...
where A(x - tanh(x^2)) = x and A(x) = x + tanh(A(x)^2).
Series expansions:
A(x) = x + tanh(x^2) + d/dx tanh(x^2)^2/2! + d^2/dx^2 tanh(x^2)^3/3! + d^3/dx^3 tanh(x^2)^4/4! +...
log(A(x)/x) = tanh(x^2)/x + d/dx (tanh(x^2)^2/x)/2! + d^2/dx^2 (tanh(x^2)^3/x)/3! + d^3/dx^3 (tanh(x^2)^4/x)/4! +...

Mathematica

Rest[CoefficientList[InverseSeries[Series[x - Tanh[x^2], {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 23 2014 *)

Prog

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

Crossrefs

Cf. A226759, A215188, A226758.

Sequence in context: A245067 A052680 A096317 * A226758 A214770 A081470

Adjacent sequences: A226757 A226758 A226759 * A226761 A226762 A226763

Keyword

nonn

Author

Paul D. Hanna, Jun 16 2013

Status

approved