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%I #2 Mar 30 2012 18:37:16
%S 1,1,-2,0,26,0,-1378,0,141202,0,-22716418,0,5218302090,0,
%T -1619288968386,0,653379470919714,0,-333014944014777730,0,
%U 209463165121436380282,0,-159492000935562428176162,0,144654795258284936534929586,0
%N G.f. A(x) satisfies the condition that both A(x) and F(x) = A(x*F(x)^2) have zeros for every other coefficient after initial terms; dual sequence A157302 satisfies the same condition.
%F For n>=2, [x^(2n)] A(x)^(4n+1) = 0.
%F G.f. satisfies: A(x) = F(x/A(x)^2) where F(x) = A(x*F(x)^2) = sqrt(Series_Reversion(x/A(x)^2)/x) = g.f. of A157307.
%F G.f. satisfies: A(x) = G(x/A(x)) where G(x) = A(x*G(x)) = Series_Reversion(x/A(x))/x = g.f. of A157306.
%e G.f.: A(x) = 1 + x - 2*x^2 + 26*x^4 - 1378*x^6 + 141202*x^8 -+...
%e ...
%e Let F(x) = A(x*F(x)^2) so that A(x) = F(x/A(x)^2) then
%e F(x) = 1 + x - 7*x^3 + 242*x^5 - 17771*x^7 + 2189294*x^9 -+...
%e has alternating zeros in the coefficients (cf. A157304):
%e [1,1,0,-7,0,242,0,-17771,0,2189294,0,-404590470,0,104785114020,0,...].
%e ...
%e COEFFICIENTS IN ODD POWERS OF G.F. A(x).
%e A^1: [(1),1,-2,0,26,0,-1378,0,141202,0,-22716418,0,...];
%e A^3: [1,(3),-3,-11,84,168,-4376,-8580,438348,865776,...];
%e A^5: [1,5,(0),-30,115,601,-7120,-30280,726680,2987400,...];
%e A^7: [1,7,7,(-49),91,1253,-8743,-65519,964768,6410880,...];
%e A^9: [1,9,18,-60,(0),1998,-8418,-112284,1106775,11070241,...];
%e A^11:[1,11,33,-55,-154,(2662),-5566,-166034,1108657,...];
%e A^13:[1,13,52,-26,-351,3055,(0),-220116,935051,23169939,...];
%e A^15:[1,15,75,35,-555,3003,7995,(-266565),565635,29818365,...];
%e A^17:[1,17,102,136,-714,2380,17646,-297160,(0),36161142,...];
%e A^19:[1,19,133,285,-760,1140,27740,-304608,-739670,(41596586),...];
%e ...
%e When scaled, the coefficients shown above in parenthesis
%e forms the coefficients of the function F(x) = A(x*F(x)^2):
%e F: [1,3/3,0,-49/7,0,2662/11,0,-266565/15,0,41596586/19,0,...].
%o (PARI) {a(n)=local(A=[1, 1]); for(i=1, n, if(#A%2==0, A=concat(A, t); A[ #A]=-subst(Vec(serreverse(x/Ser(A)))[ #A], t, 0)); if(#A%2==1, A=concat(A, t); A[ #A]=-subst(Vec(x/serreverse(x*Ser(A)))[ #A], t, 0))); Vec(x/serreverse(x*Ser(A)))[n+1]}
%Y Cf. A157306, A157307, A157302 (dual), A157303, A157304.
%K sign
%O 0,3
%A _Paul D. Hanna_, Feb 28 2009
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