A214224 E.g.f. satisfies: A(x) = x/(1 - tan(A(x))).

1, 2, 12, 128, 1920, 37056, 874496, 24395776, 785387520, 28658483200, 1168842842112, 52692107132928, 2601710674640896, 139635670319366144, 8094064830515773440, 503939620849307353088, 33539757103898142179328, 2376284247629812872511488, 178564437032337539449487360

Offset

1,2

Links

Robert Israel, Table of n, a(n) for n = 1..362

Formula

E.g.f. A(x) satisfies:

(1) A(x - x*tan(x)) = x.

(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n*tan(x)^n/n!.

(3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1)*tan(x)^n/n! ).

a(n) = n*A201594(n-1).

a(n) = (n-1)! * [x^n] x/(1 - tan(x))^n.

Example

E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 128*x^4/4! + 1920*x^5/5! +...
Related expansions:
A(x) = x + x*tan(x) + d/dx x^2*tan(x)^2/2! + d^2/dx^2 x^3*tan(x)^3/3! + d^3/dx^3 x^4*tan(x)^4/4! +...
log(A(x)/x) = tan(x) + d/dx x*tan(x)^2/2! + d^2/dx^2 x^2*tan(x)^3/3! + d^3/dx^3 x^3*tan(x)^4/4! +...
A(x)/x = 1 + x + 4*x^2/2! + 32*x^3/3! + 384*x^4/4! + 6176*x^5/5! + 124928*x^6/6! +...+ A201594(n)*x^n/n! +...
tan(A(x)) = x + 2*x^2/2! + 14*x^3/3! + 152*x^4/4! + 2296*x^5/5! + 44496*x^6/6! + 1052848*x^7/7! + 29425024*x^8/8! +...

Maple

f:= b*(1-tan(b))-x:
newt:= unapply(b-normal(f/diff(f, b)), b):
B:= x:
for n from 1 to 5 do
  B:= convert(series(newt(B), x, 2^n+1), polynom)
od:
seq(coeff(B, x, j)*j!, j=1..2^5); # Robert Israel, Feb 04 2019

Mathematica

m = 20; CoefficientList[InverseSeries[Series[x(1-Tan[x]), {x, 0, m}], x]/x, x] Range[m]! (* Jean-François Alcover, Apr 01 2019 *)

Prog

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

Crossrefs

Cf. A201594, A214222, A214223, A214225.

Sequence in context: A366821 A003712 A143136 * A214431 A227461 A367374

Adjacent sequences: A214221 A214222 A214223 * A214225 A214226 A214227

Keyword

nonn

Author

Paul D. Hanna, Jul 07 2012

Status

approved