OFFSET
1,2
FORMULA
E.g.f.: Series_Reversion( x - tan(x)^2 ).
E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) tan(x)^(2*n)/n!.
E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (tan(x)^(2*n)/x)/n! ).
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
EXAMPLE
E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 136*x^4/4! + 2160*x^5/5! +...
where A(x - tan(x)^2) = x.
Related expansions:
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! +...
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! +...
tan(A(x)) = x + 2*x^2/2! + 14*x^3/3! + 160*x^4/4! + 2536*x^5/5! + 51632*x^6/6! +...
tan(A(x))^2 = 2*x^2/2! + 12*x^3/3! + 136*x^4/4! + 2160*x^5/5! +...
tan(x) = x + 2*x^3/3! + 16*x^5/5! + 272*x^7/7! + 7936*x^9/9! +...
tan(x)^2 = 2*x^2/2! + 16*x^4/4! + 272*x^6/6! + 7936*x^8/8! +...
MATHEMATICA
Rest[CoefficientList[InverseSeries[Series[x - Tan[x]^2, {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 12 2014 *)
PROG
(PARI) a(n, m=1)=n!*polcoeff(serreverse(x-tan(x+x*O(x^n))^2), 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, tan(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, tan(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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 02 2011
STATUS
approved