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%I #6 May 22 2015 17:44:53
%S 1,2,0,0,2,-4,0,0,-16,40,0,0,200,-544,0,0,-3006,8540,0,0,49956,
%T -145720,0,0,-884352,2625648,0,0,16349648,-49161024,0,0,-311986480,
%U 947069352,0,0,6098614912,-18650752400,0,0,-121497078016,373773754912,0,0
%N G.f. A(x) satisfies: A(x)^2 = 1/AGM(1, 1 - 8*x/A(x)^2 ).
%C See A060691 for the expansion of AGM(1,1-8x), where AGM denotes the arithmetic-geometric mean.
%F G.f.: A(x) = sqrt( x/Series_Reversion( x/AGM(1,1-8*x) ) ).
%F Self-convolution equals A158100.
%F Contribution from _Paul D. Hanna_, Mar 14 2009: (Start)
%F Quadrasections are A158212(n) = A158122(4n) and A158213 = A158122(4n+1);
%F let B(x), C(x), be the g.f.s of A158212 and A158213, respectively,
%F then C(x) = 2/B(x) so that
%F A(x) = B(x^4) + x*C(x^4) = B(x^4) + 2*x/B(x^4) = 2/C(x^4) + x*C(x^4). (End)
%e G.f.: A(x) = 1 + 2*x + 2*x^4 - 4*x^5 - 16*x^8 + 40*x^9 + 200*x^12 -+...
%e A(x)^2 = 1 + 4*x + 4*x^2 + 4*x^4 - 16*x^6 - 28*x^8 + 176*x^10 +...
%e Contribution from _Paul D. Hanna_, Mar 14 2009: (Start)
%e G.f. of quadrasection A158212 is:
%e B(x) = 1 + 2*x - 16*x^2 + 200*x^3 - 3006*x^4 + 49956*x^5 +...;
%e G.f. of quadrasection A158213 is C(x) = 2/B(x):
%e C(x) = 2 - 4*x + 40*x^2 - 544*x^3 + 8540*x^4 - 145720*x^5 +...
%e where g.f. A(x) = B(x^4) + x*C(x^4) = B(x^4) + 2*x/B(x^4). (End)
%o (PARI) {a(n)=polcoeff(sqrt(x/serreverse(x/agm(1,1-8*x +x*O(x^n)))),n)}
%Y Cf. A060691, A158100 (self-convolution), A258053.
%Y Cf. quadrasections: A158212, A158213.
%K sign
%O 0,2
%A _Paul D. Hanna_, Mar 13 2009
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