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A361778 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} x^n * ((-x)^(n-1) - 2*A(x))^n.

%I A361778 #27 May 12 2023 06:35:54
%S A361778 1,2,7,27,109,459,2006,9017,41384,193048,912571,4361939,21045710,
%T A361778 102361864,501349447,2470556294,12240270901,60935582862,304660949343,
%U A361778 1529125824203,7701783889261,38915600049447,197206343307012,1002023916642621,5103911800972155,26056404563941575
%N A361778 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} x^n * ((-x)^(n-1) - 2*A(x))^n.
%H A361778 Paul D. Hanna, <a href="/A361778/b361778.txt">Table of n, a(n) for n = 0..300</a>
%F A361778 G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
%F A361778 (1) 1 = Sum_{n=-oo..+oo} x^n * ((-x)^(n-1) - 2*A(x))^n.
%F A361778 (2) 1 = Sum_{n=-oo..+oo} x^(2*n+1) * (2*A(x) - (-x)^n)^n.
%F A361778 (3) 2*A(x) = Sum_{n=-oo..+oo} x^(3*n+1) * ((-x)^n - 2*A(x))^n.
%F A361778 (4) 1 = Sum_{n=-oo..+oo} x^(n^2) / (1 - 2*A(x)*(-x)^(n+1))^n.
%F A361778 (5) 1 = Sum_{n=-oo..+oo} x^(n^2) / (1 - 2*A(x)*(-x)^(n+1))^(n+1).
%F A361778 (6) 2*A(x) = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 - 2*A(x)*(-x)^(n+1))^(n+1).
%F A361778 (7) 0 = Sum_{n=-oo..+oo} x^(2*n) * (2*A(x) - (-x)^n)^(n+1).
%F A361778 (8) 0 = Sum_{n=-oo..+oo} x^(3*n) * ((-x)^(n-1) - 2*A(x))^(n+1).
%e A361778 G.f.: A(x) = 1 + 2*x + 7*x^2 + 27*x^3 + 109*x^4 + 459*x^5 + 2006*x^6 + 9017*x^7 + 41384*x^8 + 193048*x^9 + 912571*x^10 + ...
%e A361778 SPECIFIC VALUES.
%e A361778 A(1/7) = 1.63053651133635034184414884744745628155427916612173429157...
%e A361778 A(1/6) = 1.99892384479086071017436459041327119822244448085100733509...
%e A361778 A(x) = 2 at x = 0.166713109990638926829644490786806133084979604287174064...
%e A361778 Radius of convergence r and the value A(r) are given by
%e A361778 r = 0.182033752413024354859591633469061831146023401652842514076551...
%e A361778 A(r) = 2.63999965897091399750291467200041973752650665197493948118984006...
%e A361778 1/r = 5.4934867119096473651972990947886642212447897087082048838...
%o A361778 (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
%o A361778 A[#A] = polcoeff( sum(m=-#A, #A, x^m * ((-x)^(m-1) - 2*Ser(A))^m ), #A)/2); A[n+1]}
%o A361778 for(n=0, 30, print1(a(n), ", "))
%Y A361778 Cf. A355865, A357227, A359712.
%K A361778 nonn
%O A361778 0,2
%A A361778 _Paul D. Hanna_, May 10 2023