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G.f. A(x) satisfies: x = Sum_{n>=1} (-1)^(n-1) * x^(2*n-1)*A(x) / (1 + x^(2*n-1)*A(x)).
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%I #6 Apr 20 2018 18:29:22

%S 1,1,2,4,7,13,26,50,96,190,377,747,1494,3004,6051,12237,24843,50557,

%T 103143,210975,432461,888173,1827562,3766980,7776620,16077958,

%U 33286760,69002906,143213917,297573927,618964149,1288754681,2685872873,5602584099,11696560369,24438577665,51100370596,106926690324,223896358139,469129457585

%N G.f. A(x) satisfies: x = Sum_{n>=1} (-1)^(n-1) * x^(2*n-1)*A(x) / (1 + x^(2*n-1)*A(x)).

%C Note that 1 + 4*Sum_{n>=1} (-1)^n * x^(2*n-1)/(1 + x^(2*n-1)) = theta_4(x)^2, where theta_4(x) = 1 + 2*Sum_{n>=1} (-x)^(n^2) is Jacobi's elliptic theta function.

%H Paul D. Hanna, <a href="/A303059/b303059.txt">Table of n, a(n) for n = 0..900</a>

%e G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 7*x^4 + 13*x^5 + 26*x^6 + 50*x^7 + 96*x^8 + 190*x^9 + 377*x^10 + 747*x^11 + 1494*x^12 + ...

%e such that

%e x = x*A(x)/(1 + x*A(x)) - x^3*A(x)/(1 + x^3*A(x)) + x^5*A(x)/(1 + x^5*A(x)) - x^7*A(x)/(1 + x^7*A(x)) + x^9*A(x)/(1 + x^9*A(x)) -+ ...

%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0); A[#A] = Vec( sum(m=1,#A,(-1)^m*x^(2*m-1)*Ser(A)/(1+x^(2*m-1)*Ser(A) )) )[#A] );A[n+1]}

%o for(n=0,60,print1(a(n),", "))

%K nonn

%O 0,3

%A _Paul D. Hanna_, Apr 20 2018