A214224 - OEIS

%I #12 Apr 01 2019 07:22:49

%S 1,2,12,128,1920,37056,874496,24395776,785387520,28658483200,

%T 1168842842112,52692107132928,2601710674640896,139635670319366144,

%U 8094064830515773440,503939620849307353088,33539757103898142179328,2376284247629812872511488,178564437032337539449487360

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

%H Robert Israel, <a href="/A214224/b214224.txt">Table of n, a(n) for n = 1..362</a>

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

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

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

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

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

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

%e E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 128*x^4/4! + 1920*x^5/5! +...

%e Related expansions:

%e 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! +...

%e 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! +...

%e 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! +...

%e 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! +...

%p f:= b*(1-tan(b))-x:

%p newt:= unapply(b-normal(f/diff(f,b)),b):

%p B:= x:

%p for n from 1 to 5 do

%p   B:= convert(series(newt(B),x,2^n+1),polynom)

%p od:

%p seq(coeff(B,x,j)*j!, j=1..2^5); # _Robert Israel_, Feb 04 2019

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

%o (PARI) {a(n)=(n-1)!*polcoeff(x/(1 - tan(x+x*O(x^n)))^n,n)}

%o for(n=1, 25, print1(a(n), ", "))

%o (PARI) {a(n)=n!*polcoeff(serreverse(x-x*tan(x+x*O(x^n))), n)}

%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}

%o {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)}

%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}

%o {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)}

%Y Cf. A201594, A214222, A214223, A214225.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jul 07 2012