A226760 - OEIS

%I #5 Jan 23 2014 10:05:36

%S 1,2,12,120,1680,30000,651840,16692480,492307200,16432899840,

%T 612414190080,25205181281280,1135434972672000,55567263749437440,

%U 2935699697464934400,166525788640142131200,10094443872515806003200,651211844046521735577600,44547029929095149430374400

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

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

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

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

%F 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

%e E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 120*x^4/4! + 1680*x^5/4! +...

%e where A(x - tanh(x^2)) = x and A(x) = x + tanh(A(x)^2).

%e Series expansions:

%e 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! +...

%e 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! +...

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

%o (PARI) {a(n)=n!*polcoeff(serreverse(x-tanh(x^2+x^2*O(x^n))), n)}

%o for(n=1, 25, print1(a(n), ", "))

%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}

%o {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)}

%o for(n=1, 25, print1(a(n), ", "))

%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}

%o {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)}

%o for(n=1, 25, print1(a(n), ", "))

%Y Cf. A226759, A215188, A226758.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jun 16 2013