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
Keywords
Examples
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! +...
Links
- Robert Israel, Table of n, a(n) for n = 1..362
Programs
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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
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Mathematica
m = 20; CoefficientList[InverseSeries[Series[x(1-Tan[x]), {x, 0, m}], x]/x, x] Range[m]! (* Jean-François Alcover, Apr 01 2019 *) -
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)}
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.