%I #13 Jan 12 2014 11:17:12
%S 1,1,5,44,581,10256,227529,6088256,190930729,6870227200,279066777613,
%T 12632667642880,630670054092525,34426087332253696,2039903110075608977,
%U 130404672744539242496,8946117466489960168913,655585000075494566199296,51111210765059412626238741
%N E.g.f. satisfies: A(x) = 1/(cos(x*A(x)) - sin(x*A(x))).
%C Compare e.g.f. to: LambertW(-x)/(-x) = (1/x)*Series_Reversion(x*(cosh(x) - sinh(x))).
%C The radius of convergence r of e.g.f. A(x) is given by:
%C r = t*(cos(t) - sin(t)) where tan(t) = (1-t)/(1+t), which evaluates as:
%C r = 0.21266685344074710045360679397024815598865409988038...
%C t = 0.40262817418811160981993252391123072456350647779608...
%C Further, A(r) = 1/(cos(t) - sin(t)), thus
%C A(r) = 1.89323426605496483543109751303457163422769666683274...
%F E.g.f. satisfies: A( x*(cos(x) - sin(x)) ) = 1/(cos(x) - sin(x)).
%F E.g.f: (1/x) * Series_Reversion( x*(cos(x) - sin(x)) ).
%F a(n) = [x^n/n!] 1/(cos(x)-sin(x))^(n+1) / (n+1).
%F a(n) ~ n^(n-1) * sqrt((t*cos(2*t))/(3+sin(2*t))) / (exp(n) * r^(n+1)), where r and t were described above. - _Vaclav Kotesovec_, Jan 12 2014
%e E.g.f.: A(x) = 1 + x + 5*x^2/2! + 44*x^3/3! + 581*x^4/4! + 10256*x^5/5! +...
%e where
%e 1/(cos(x)-sin(x)) = 1 + x + 3*x^2/2! + 11*x^3/3! + 57*x^4/4! + 361*x^5/5! + 2763*x^6/6! +...+ A001586(n)*x^n/n! +...
%e The coefficient of x^n/n! in powers of G(x) = 1/(cos(x)-sin(x)) begins:
%e G^1: [(1), 1, 3, 11, 57, 361, 2763, 24611, ..., A001586(n), ...];
%e G^2: [1,(2), 8, 40, 256, 1952, 17408, 177280, ..., A000828(n+1), ...];
%e G^3: [1, 3,(15), 93, 705, 6243, 63375, 724413, ...];
%e G^4: [1, 4, 24,(176), 1536, 15424, 175104, 2214656, ...];
%e G^5: [1, 5, 35, 295,(2905), 32525, 407435, 5638495, ...];
%e G^6: [1, 6, 48, 456, 4992, (61536), 841728, 12633216, ...];
%e G^7: [1, 7, 63, 665, 8001, 107527, (1592703), 25738265, ...];
%e G^8: [1, 8, 80, 928, 12160, 176768, 2816000, (48706048), ...]; ...
%e where coefficients in parenthesis form the initial terms of this sequence:
%e [1/1, 2/2, 15/3, 176/4, 2905/5, 61536/6, 1592703/7, 48706048/8, ...].
%t CoefficientList[1/x*InverseSeries[Series[x*(Cos[x] - Sin[x]), {x, 0, 21}], x],x] * Range[0, 20]! (* _Vaclav Kotesovec_, Jan 12 2014 *)
%o (PARI) {a(n)=local(X=x+x*O(x^n));n!*polcoeff(1/x*serreverse(x*(cos(X)-sin(X) )),n)}
%o (PARI) {a(n)=local(A=1+x,X=x+x*O(x^n));for(i=1,n,A=1/(cos(X*A) - sin(X*A)));n!*polcoeff(A,n)}
%Y Cf. A001586.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Dec 06 2011