cp's OEIS Frontend

A201561 E.g.f. satisfies: A(x) = x + tan( A(x) )^2 with A(0)=0.

%I A201561 #12 Jan 12 2014 11:17:01
%S A201561 1,2,12,136,2160,43952,1092672,32102656,1088252160,41809041152,
%T A201561 1795201638912,85196352787456,4428299422310400,250187205957220352,
%U A201561 15265712890413023232,1000468694343925006336,70089639485229413498880,5227049493330884279140352,413441163603081566484037632
%N A201561 E.g.f. satisfies: A(x) = x + tan( A(x) )^2 with A(0)=0.
%F A201561 E.g.f.: Series_Reversion( x - tan(x)^2 ).
%F A201561 E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) tan(x)^(2*n)/n!.
%F A201561 E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (tan(x)^(2*n)/x)/n! ).
%F A201561 a(n) ~ t*sqrt(((1+arccos(t))*t^2-1)/(6-4*t^2)) * n^(n-1) / (exp(n) * (1+arccos(t)-1/t^2)^n), where t = sqrt(((6*(9+sqrt(129)))^(1/3) - 2*6^(2/3)/(9+sqrt(129))^(1/3))/3) = 0.920710376904467468... is the root of the equation 4-4*t^2 = t^6. - _Vaclav Kotesovec_, Jan 12 2014
%e A201561 E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 136*x^4/4! + 2160*x^5/5! +...
%e A201561 where A(x - tan(x)^2) = x.
%e A201561 Related expansions:
%e A201561 A(x) = x + tan(x)^2 + d/dx tan(x)^4/2! + d^2/dx^2 tan(x)^6/3! + d^3/dx^3 tan(x)^8/4! +...
%e A201561 log(A(x)/x) = tan(x)^2/x + d/dx (tan(x)^4/x)/2! + d^2/dx^2 (tan(x)^6/x)/3! + d^3/dx^3 (tan(x)^8/x)/4! +...
%e A201561 tan(A(x)) = x + 2*x^2/2! + 14*x^3/3! + 160*x^4/4! + 2536*x^5/5! + 51632*x^6/6! +...
%e A201561 tan(A(x))^2 = 2*x^2/2! + 12*x^3/3! + 136*x^4/4! + 2160*x^5/5! +...
%e A201561 tan(x) = x + 2*x^3/3! + 16*x^5/5! + 272*x^7/7! + 7936*x^9/9! +...
%e A201561 tan(x)^2 = 2*x^2/2! + 16*x^4/4! + 272*x^6/6! + 7936*x^8/8! +...
%t A201561 Rest[CoefficientList[InverseSeries[Series[x - Tan[x]^2, {x, 0, 20}], x],x] * Range[0, 20]!] (* _Vaclav Kotesovec_, Jan 12 2014 *)
%o A201561 (PARI) a(n, m=1)=n!*polcoeff(serreverse(x-tan(x+x*O(x^n))^2),n)
%o A201561 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A201561 {a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, tan(x+x*O(x^n))^(2*m)/m!)); n!*polcoeff(A, n)}
%o A201561 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A201561 {a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, tan(x+x*O(x^n))^(2*m)/x/m!)+x*O(x^n))); n!*polcoeff(A, n)}
%o A201561 for(n=1, 25, print1(a(n), ", "))
%Y A201561 Cf. A205886.
%K A201561 nonn
%O A201561 1,2
%A A201561 _Paul D. Hanna_, Dec 02 2011