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%I #2 Mar 30 2012 18:37:16
%S 1,1,0,-5,0,183,0,-14352,0,1857199,0,-355082433,0,94134281460,0,
%T -33120720127500,0,14959943533260783,0,-8447188671812872887,0,
%U 5834800994047642310223,0,-4842259038722174600622240,0
%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; g.f. of dual sequence A157305 satisfies the same condition.
%F For n>=2, [x^(2n-1)] 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 A157304.
%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 A157303.
%e G.f.: A(x) = 1 + x - 5*x^3 + 183*x^5 - 14352*x^7 + 1857199*x^9 -+...
%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 + 2*x^2 - 26*x^4 + 1378*x^6 - 141202*x^8 +-...
%e has alternating zeros in the coefficients (cf. A157304):
%e [1,1,2,0,-26,0,1378,0,-141202,0,22716418,0,-5218302090,0,...].
%e ...
%e COEFFICIENTS IN ODD POWERS OF G.F. A(x).
%e A^1: [(1),1,0,-5,0,183,0,-14352,0,1857199,0,...];
%e A^3: [1,(3),3,-14,-30,534,1173,-42432,-91602,5522926,...];
%e A^5: [1,5,(10),-15,-95,766,3810,-65545,-300930,8800450,...];
%e A^7: [1,7,21,(0),-175,777,7518,-79148,-610554,11321338,...];
%e A^9: [1,9,36,39,(-234),513,11640,-79866,-990603,...];
%e A^11:[1,11,55,110,-220,(0),15367,-66132,-1402005,...];
%e A^13:[1,13,78,221,-65,-624,(17914),-38571,-1801215,...];
%e A^15:[1,15,105,380,315,-1077,18760,(0),-2145855,...];
%e A^17:[1,17,136,595,1020,-901,17952,45084,(-2400434),...];
%e A^19:[1,19,171,874,2166,570,16473,91656,-2541060,(0),...];
%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: [1,3/3,10/5,0,-234/9,0,17914/13,0,-2400434/17,0,...].
%o (PARI) {a(n)=local(A=[1, 1]); for(i=1, n, if(#A%2==1, A=concat(A, t); A[ #A]=-subst(Vec(serreverse(x/Ser(A)))[ #A], t, 0)); if(#A%2==0, 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. A157303, A157304, A157305 (dual), A157306, A157307.
%K sign
%O 0,4
%A _Paul D. Hanna_, Feb 28 2009
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