A201594 - OEIS

%I #19 Jan 12 2014 11:17:05

%S 1,1,4,32,384,6176,124928,3049472,87265280,2865848320,106258440192,

%T 4391008927744,200131590356992,9973976451383296,539604322034384896,

%U 31496226303081709568,1972926888464596598784,132015791534989604028416,9398128264859870497341440,709248762402156849800413184

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

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

%F E.g.f.: (1/x)*Series_Reversion( x*(1-tan(x)) ).

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

%F a(n) = A214224(n+1)/(n+1).

%F a(n) ~ n^(n-1) * ((t^2+1)/(t-1)^2)^(n+1/2) / (sqrt(2*(t+1)) * exp(n)), where t = 0.46733877379062994365... is the root of the equation t = tan((1-t)/(1+t^2)). - _Vaclav Kotesovec_, Jan 12 2014

%e E.g.f.: A(x) = 1 + x + 4*x^2/2! + 32*x^3/3! + 384*x^4/4! + 6176*x^5/5! +...

%e The coefficients in the initial powers of G(x) = 1/(1 - tan(x)) begin:

%e G^1: [(1), 1, 2, 8, 40, 256, 1952, 17408, ..., A000828(n), ...];

%e G^2: [1,(2), 6, 28, 168, 1232, 10656, 106048, ...];

%e G^3: [1, 3,(12), 66, 456, 3768, 36192, 395616, ...];

%e G^4: [1, 4, 20,(128), 1000, 9184, 96800, 1150208, ...];

%e G^5: [1, 5, 30, 220,(1920), 19400, 222480, 2852320, ...];

%e G^6: [1, 6, 42, 348, 3360,(37056), 459312, 6317088, ...];

%e G^7: [1, 7, 56, 518, 5488, 65632, (874496), 12841808, ...];

%e G^8: [1, 8, 72, 736, 8496, 109568, 1562112, (24395776), ...]; ...

%e where coefficients in parenthesis form initial terms of this sequence:

%e [1/1, 2/2, 12/3, 128/4, 1920/5, 37056/6, 874496/7, 24395776/8, ...].

%t CoefficientList[1/x*InverseSeries[Series[x*(1-Tan[x]), {x, 0, 21}], x],x] * Range[0, 20]! (* _Vaclav Kotesovec_, Jan 12 2014 *)

%o (PARI) {a(n)=n!*polcoeff(1/x*serreverse(x-x*tan(x+x^2*O(x^n))),n)}

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

%Y Cf. A214224, A201595, A201128, A000828.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 02 2011