cp's OEIS Frontend

A226759 E.g.f.: A(x) = x + tan(A(x)^2).

%I A226759 #7 Jan 23 2014 10:03:45
%S A226759 1,2,12,120,1680,30480,678720,17902080,545529600,18854519040,
%T A226759 728651911680,31133305082880,1457247407616000,74151941277173760,
%U A226759 4075563460173004800,240617659203765043200,15186689706926068531200,1020415122190724766105600,72722026905140804154163200
%N A226759 E.g.f.: A(x) = x + tan(A(x)^2).
%F A226759 E.g.f.: Series_Reversion(x - tan(x^2)).
%F A226759 E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) tan(x^2)^n/n!.
%F A226759 E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (1/x)*tan(x^2)^n/n! ).
%F A226759 a(n) ~ n^(n-1) * sqrt(r*s/(1 + 4*s^2*(s-r))) / (exp(n) * r^n), where s = 0.4749798472710374904... is the root of the equation 2*s = cos(s^2)^2, and r = s - tan(s^2) = 0.2454667961619663296... - _Vaclav Kotesovec_, Jan 23 2014
%e A226759 E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 120*x^4/4! + 1680*x^5/4! +...
%e A226759 where A(x - tan(x^2)) = x and A(x) = x + tan(A(x)^2).
%e A226759 Series expansions:
%e A226759 A(x) = x + tan(x^2) + d/dx tan(x^2)^2/2! + d^2/dx^2 tan(x^2)^3/3! + d^3/dx^3 tan(x^2)^4/4! +...
%e A226759 log(A(x)/x) = tan(x^2)/x + d/dx (tan(x^2)^2/x)/2! + d^2/dx^2 (tan(x^2)^3/x)/3! + d^3/dx^3 (tan(x^2)^4/x)/4! +...
%t A226759 Rest[CoefficientList[InverseSeries[Series[x - Tan[x^2],{x,0,20}],x],x] * Range[0,20]!] (* _Vaclav Kotesovec_, Jan 23 2014 *)
%o A226759 (PARI) {a(n)=n!*polcoeff(serreverse(x-tan(x^2+x^2*O(x^n))), n)}
%o A226759 for(n=1, 25, print1(a(n), ", "))
%o A226759 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A226759 {a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, tan(x^2+x*O(x^n))^m)/m!); n!*polcoeff(A, n)}
%o A226759 for(n=1, 25, print1(a(n), ", "))
%o A226759 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A226759 {a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, tan(x^2+x*O(x^n))^m/x)/m!)+x*O(x^n)); n!*polcoeff(A, n)}
%o A226759 for(n=1, 25, print1(a(n), ", "))
%Y A226759 Cf. A226760, A215188, A226758.
%K A226759 nonn
%O A226759 1,2
%A A226759 _Paul D. Hanna_, Jun 16 2013