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A193618 G.f. A(x) satisfies: A(x)^2 + A(-x)^2 = 2 and A(x)^-2 - A(-x)^-2 = -8*x.

%I A193618 #15 Aug 26 2020 01:53:15
%S A193618 1,2,-2,-28,54,860,-2004,-33720,86054,1492908,-4019452,-71101832,
%T A193618 198310460,3555617432,-10168382696,-184127171952,536496907782,
%U A193618 9788598556876,-28937139277804,-531135371147368,1588378827366868,29295861148032584
%N A193618 G.f. A(x) satisfies: A(x)^2 + A(-x)^2 = 2 and A(x)^-2 - A(-x)^-2 = -8*x.
%C A193618 The unsigned version of this sequence, A246062, has g.f.: sqrt( (1 + sqrt(1+8*x)) / (1 + sqrt(1-8*x)) ).
%H A193618 Seiichi Manyama, <a href="/A193618/b193618.txt">Table of n, a(n) for n = 0..1000</a>
%F A193618 G.f.: ( 2*(sqrt(1+64*x^2) + 8*x)/(sqrt(1+64*x^2) + 1) )^(1/4).
%F A193618 G.f. A(x) = 1/G(x) where G(x) is the g.f. of A193619.
%e A193618 G.f.: A(x) = 1 + 2*x - 2*x^2 - 28*x^3 + 54*x^4 + 860*x^5 - 2004*x^6 +...
%e A193618 where
%e A193618 A(x)^2 = 1 + 4*x - 64*x^3 + 2048*x^5 - 81920*x^7 + 3670016*x^9 +...
%e A193618 and
%e A193618 A(x)^-2 = 1 - 4*x + 16*x^2 - 256*x^4 + 8192*x^6 - 327680*x^8 +...
%o A193618 (PARI) {a(n)=local(Ox=x*O(x^n),A=(2*(sqrt(1+64*x^2+Ox)+8*x)/(sqrt(1+64*x^2+Ox)+1))^(1/4));polcoeff(A,n)}
%o A193618 (PARI) N=40; x='x+O('x^N); Vec(sqrt(2/(1-8*x+sqrt(1+64*x^2)))) \\ _Seiichi Manyama_, Aug 26 2020
%Y A193618 Cf. A193619, A246062.
%K A193618 sign
%O A193618 0,2
%A A193618 _Paul D. Hanna_, Aug 01 2011