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EXAMPLE
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G.f.: A(x) = 1 + x + 3*x^2 + 38*x^3 + 947*x^4 + 37394*x^5 +...
RELATED FUNCTIONS.
G.f. of A157308, B(x) = x + A(-x^2), satisfies the condition
that both B(x) and F(x) = B(x*F(x)) = o.g.f. of A155585
have zeros for every other coefficient after initial terms:
A157308 = [1,1,-1,0,3,0,-38,0,947,0,-37394,0,2120190,0,...];
A155585 = [1,1,0,-2,0,16,0,-272,0,7936,0,-353792,0,...].
...
G.f. of A157310, C(x) = 2+x - A(-x^2), satisfies the condition
that both C(x) and G(x) = C(x/G(x)) = o.g.f. of A157309
have zeros for every other coefficient after initial terms:
A157310 = [1,1,1,0,-3,0,38,0,-947,0,37394,0,-2120190,0,...];
A157309 = [1,1,0,-1,0,9,0,-176,0,5693,0,-272185,0,...].
...
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PROGRAM
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(PARI) {a(n)=local(A=[1, 1]); for(i=1, 2*n, if(#A%2==0, A=concat(A, 0); ); if(#A%2==1, A=concat(A, t); A[ #A]=-subst(Vec(x/serreverse(x*Ser(A)))[ #A], t, 0))); (-1)^n*Vec(x/serreverse(x*Ser(A)))[2*n+1]}
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