This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A157307 #2 Mar 30 2012 18:37:16
%S A157307 1,1,0,-7,0,242,0,-17771,0,2189294,0,-404590470,0,104785114020,0,
%T A157307 -36267349272243,0,16185534555969878,0,-9056084951164614194,0,
%U A157307 6210575966204050489916,0,-5124228589785627978616702,0
%N A157307 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 A157304 satisfies the same condition.
%F A157307 For n>=1, [x^(2n+1)] 1/A(x)^(4n+1) = 0.
%F A157307 G.f. satisfies: A(x) = F(x*A(x)^2) where F(x) = A(x/F(x)^2) = sqrt(x/Series_Reversion(x*A(x)^2)) = g.f. of A157305.
%F A157307 G.f. satisfies: A(x) = G(x*A(x)) where G(x) = A(x/G(x)) = x/Series_Reversion(x*A(x)) = g.f. of A157306.
%e A157307 G.f.: A(x) = 1 + x - 7*x^3 + 242*x^5 - 17771*x^7 + 2189294*x^9 -+...
%e A157307 ...
%e A157307 Let F(x) = A(x/F(x)^2) so that A(x) = F(x*A(x)^2) then
%e A157307 F(x) = 1 + x - 2*x^2 + 26*x^4 - 1378*x^6 + 141202*x^8 -+...
%e A157307 has alternating zeros in the coefficients (cf. A157305):
%e A157307 [1,1,-2,0,26,0,-1378,0,141202,0,-22716418,0,5218302090,0,...].
%e A157307 ...
%e A157307 COEFFICIENTS IN ODD NEGATIVE POWERS OF G.F. A(x).
%e A157307 A^1 : [(1),1,0,-7,0,242,0,-17771,0,2189294,0,-404590470,0,...];
%e A157307 A^-1: [1,(-1),1,6,-13,-222,506,16932,-37709,-2127126,4595294,...];
%e A157307 A^-3: [1,-3,(6),11,-69,-537,2806,45282,-215781,-5963673,...];
%e A157307 A^-5: [1,-5,15,(0),-140,-601,6245,62380,-503935,-8911515,...];
%e A157307 A^-7: [1,-7,28,-35,(-182),-392,9968,65519,-860825,-10670499,...];
%e A157307 A^-9: [1,-9,45,-102,-135,(0),13128,54504,-1240416,-11070241,...];
%e A157307 A^-11:[1,-11,66,-209,77,341,(15158),31460,-1598696,-10074240,...];
%e A157307 A^-13:[1,-13,91,-364,546,221,16107,(0),-1899508,-7767240,...];
%e A157307 A^-15:[1,-15,120,-575,1380,-978,17040,-36375,(-2118030),...];
%e A157307 A^-17:[1,-17,153,-850,2703,-4114,20502,-76772,-2240175,(0),...];
%e A157307 ...
%e A157307 When scaled, the coefficients shown above in parenthesis
%e A157307 forms the coefficients of the function F(x) = A(x/F(x)^2):
%e A157307 F: [1,-1/(-1),6/(-3),0,-182/(-7),0,15158/(-11),0,-2118030/(-15),0,...].
%o A157307 (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(serreverse(x/Ser(A))/x)[n+1]}
%Y A157307 Cf. A157305, A157306, A157302, A157303, A157304 (dual).
%K A157307 sign
%O A157307 0,4
%A A157307 _Paul D. Hanna_, Feb 28 2009