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A157309 G.f. A(x) satisfies the condition that both A(x) and F(x) = A(x*F(x)) = o.g.f. of A157310 have zeros for every other coefficient after initial terms; g.f. of dual sequence A157308 satisfies the same condition.

%I A157309 #2 Mar 30 2012 18:37:16
%S A157309 1,1,0,-1,0,9,0,-176,0,5693,0,-272185,0,18043492,0,-1587355800,0,
%T A157309 179258676373,0,-25305967691715,0,4370075849887361,0,
%U A157309 -906689353191842372,0,222613537277330398444,0,-63850898347335510126988
%N A157309 G.f. A(x) satisfies the condition that both A(x) and F(x) = A(x*F(x)) = o.g.f. of A157310 have zeros for every other coefficient after initial terms; g.f. of dual sequence A157308 satisfies the same condition.
%F A157309 Let F(x) = o.g.f. of A157310, then F(x) satisfies:
%F A157309 A(x) = Series_Reversion(x/F(x))/x;
%F A157309 A(x) = F(x/A(x));
%F A157309 F(x) = A(x*F(x));
%F A157309 where A157310 has zeros for every other term after initial [1,1,1].
%e A157309 G.f.: A(x) = 1 + x - x^3 + 9*x^5 - 176*x^7 + 5693*x^9 -+...
%e A157309 RELATED FUNCTIONS.
%e A157309 If F(x) = A(x*F(x)) then F(x) = o.g.f. of A157310:
%e A157309 A157310 = [1,1,1,0,-3,0,38,0,-947,0,37394,0,-2120190,0,...];
%e A157309 has zeros for every other coefficient after initial terms.
%e A157309 ...
%e A157309 O.g.f. A(x) has similar properties as o.g.f. of A157308:
%e A157309 A157308 = [1,1,-1,0,3,0,-38,0,947,0,-37394,0,2120190,0,...].
%o A157309 (PARI) {a(n)=local(A=[1, 1]); for(i=1, n, if(#A%2==1, A=concat(A, 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 A157309 Cf. A157308, A157310, A157307, A157304, A157305.
%K A157309 sign
%O A157309 0,6
%A A157309 _Paul D. Hanna_, Mar 11 2009