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A193619 G.f. A(x) satisfies: A(x)^-2 + A(-x)^-2 = 2 and A(x)^2 - A(-x)^2 = -8*x.

%I A193619 #13 Aug 26 2020 01:53:22
%S A193619 1,-2,6,12,-122,-316,4124,11608,-169018,-495724,7676596,23075112,
%T A193619 -371737956,-1135805144,18808209528,58139400112,-982459035322,
%U A193619 -3063548374604,52579900855620,165071778169864,-2868211199377740,-9053503669975944
%N A193619 G.f. A(x) satisfies: A(x)^-2 + A(-x)^-2 = 2 and A(x)^2 - A(-x)^2 = -8*x.
%H A193619 Seiichi Manyama, <a href="/A193619/b193619.txt">Table of n, a(n) for n = 0..1000</a>
%F A193619 G.f.: ( (sqrt(1+64*x^2) + 1)*(sqrt(1+64*x^2) - 8*x)/2 )^(1/4).
%F A193619 G.f. A(x) = 1/G(x) where G(x) is the g.f. of A193618.
%e A193619 G.f.: A(x) = 1 - 2*x + 6*x^2 + 12*x^3 - 122*x^4 - 316*x^5 + 4124*x^6 +...
%e A193619 where
%e A193619 A(x)^2 = 1 - 4*x + 16*x^2 - 256*x^4 + 8192*x^6 - 327680*x^8 +...
%e A193619 and
%e A193619 A(x)^-2 = 1 + 4*x - 64*x^3 + 2048*x^5 - 81920*x^7 + 3670016*x^9 +...
%o A193619 (PARI) {a(n)=local(Ox=x*O(x^n),A=((sqrt(1+64*x^2+Ox)+1)*(sqrt(1+64*x^2+Ox)-8*x)/2)^(1/4));polcoeff(A,n)}
%o A193619 (PARI) N=40; x='x+O('x^N); Vec(sqrt((1-8*x+sqrt(1+64*x^2))/2)) \\ _Seiichi Manyama_, Aug 26 2020
%Y A193619 Cf. A193618.
%K A193619 sign
%O A193619 0,2
%A A193619 _Paul D. Hanna_, Aug 01 2011