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A159601 E.g.f. S(x) satisfies: S(x) = Integral [1 - 2*S(x)^2]^(3/4) dx with S(0)=0.

%I A159601 #10 Dec 02 2013 17:31:21
%S A159601 1,-3,27,-441,11529,-442827,23444883,-1636819569,145703137041,
%T A159601 -16106380394643,2164638920874507,-347592265948756521,
%U A159601 65724760945840254489,-14454276753061349098587
%N A159601 E.g.f. S(x) satisfies: S(x) = Integral [1 - 2*S(x)^2]^(3/4) dx with S(0)=0.
%C A159601 E.g.f. S(x) is an odd function; zero terms are omitted.
%C A159601 Apart from signs and initial term, same as A159600.
%C A159601 Radius of convergence of S(x) is |x| < r where:
%C A159601 r = (1/2)*Pi^(3/2)/gamma(3/4)^2 ;
%C A159601 r = L/sqrt(2) where L=Lemniscate constant ;
%C A159601 r = 1.8540746773013719184338503471952600...
%C A159601 Although S(x) diverges at |x|=r, the power series expansion:
%C A159601 C(x) = [1 - 2*S(x)^2]^(1/4) converges to
%C A159601 C(r) = gamma(3/4)^2/(Pi/2)^(3/2) = 0.7627597635...
%F A159601 E.g.f. S(x) satisfies: C(x)^4 + 2*S(x)^2 = 1 where
%F A159601 S'(x) = C(x)^3 and C'(x) = -S(x) with C(0)=1.
%F A159601 E.g.f. S(x) satisfies: S(x)/C(x) = e.g.f. of unsigned A104203 where C(x)^4 + 2*S(x)^2 = 1.
%F A159601 a(n) = -A159600(n), n>0. - _M. F. Hasler_, Aug 31 2012
%F A159601 G.f.: (1- 1/Q(0))/x, where Q(k) = 1 + x*(2*k+1)^2/(1 + 2*x*(k+1)^2/Q(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Nov 30 2013
%e A159601 E.g.f: S(x) = x - 3*x^3/3! + 27*x^5/5! - 441*x^7/7! + 11529*x^9/9! +...
%e A159601 S(x)^2 = 2*x^2/2! - 24*x^4/4! + 504*x^6/6! - 16128*x^8/8! +-...
%e A159601 C(x)^4 + 2*S(x)^2 = 1 where:
%e A159601 C(x) = 1 - x^2/2! + 3*x^4/4! - 27*x^6/6! + 441*x^8/8! -+...
%e A159601 C(x)^2 = 1 - 2*x^2/2! + 12*x^4/4! - 144*x^6/6! + 3024*x^8/8! -+...
%e A159601 C(x)^3 = 1 - 3*x^2/2! + 27*x^4/4! - 441*x^6/6! + 11529*x^8/8! -+...
%e A159601 C(x)^4 = 1 - 4*x^2/2! + 48*x^4/4! - 1008*x^6/6! + 32256*x^8/8! -+...
%e A159601 1/C(x) = C(i*x) = 1 + x^2/2! + 3*x^4/4! + 27*x^6/6! + 441*x^8/8! +...
%e A159601 log(C(x)) = -x^2/2! - 12*x^6/6! - 3024*x^10/10! - 4390848*x^14/14! -...
%e A159601 Coefficients in log(C(x)) are given by A104203 (ignoring signs).
%o A159601 (PARI) {a(n)=local(S=x);for(i=0,2*n,S=intformal((1-2*S^2+O(x^(2*n)))^(3/4)));(2*n-1)!*polcoeff(S,2*n-1)}
%Y A159601 Cf. A159600 (C(x)), A104203 (unsigned e.g.f. = S(x)/C(x)).
%K A159601 sign
%O A159601 1,2
%A A159601 _Paul D. Hanna_, May 08 2009