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A357152 Coefficients in the power series A(x) such that: A(x)^2 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.

%I A357152 #9 Sep 16 2022 22:02:32
%S A357152 1,1,4,23,147,1022,7529,57605,453691,3653149,29937140,248865368,
%T A357152 2093488837,17787701638,152433293056,1315973808843,11434434212115,
%U A357152 99918928175263,877543565096334,7741838176253076,68576621373325887,609670801860847612,5438211584097291663
%N A357152 Coefficients in the power series A(x) such that: A(x)^2 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.
%C A357152 Related identity: 0 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1).
%H A357152 Paul D. Hanna, <a href="/A357152/b357152.txt">Table of n, a(n) for n = 0..400</a>
%F A357152 G.f. A(x) = Sum_{n>=0} a(n) * x^n satisfies the following relations.
%F A357152 (1) A(x)^2 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.
%F A357152 (2) x*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 - x^(n+1))^n * A(x)^n ).
%F A357152 (3) -x*A(x)^4 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (1 - x^(n+1)*A(x))^n.
%F A357152 (4) -A(x)^5 = Sum_{n=-oo..+oo} x^(2*n+1) * (A(x) - x^n)^(n+1) / A(x)^n.
%F A357152 (5) 0 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n*A(x))^(n+1) / A(x)^n.
%F A357152 (6) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (A(x) - x^(n+1))^n.
%e A357152 G.f.: A(x) = 1 + x + 4*x^2 + 23*x^3 + 147*x^4 + 1022*x^5 + 7529*x^6 + 57605*x^7 + 453691*x^8 + 3653149*x^9 + 29937140*x^10 + 248865368*x^11 + 2093488837*x^12 + ...
%e A357152 such that
%e A357152 A(x)^2 = ... + x^(-3)*(1 - x^(-2))^(-1)/A(x)^2 + x^(-1)/A(x) + x*0 + x^3*(1 - x)^2*A(x) + x^5*(1 - x^2)^3*A(x)^2 + x^7*(1 - x^3)^4*A(x)^3 + ... + x^(2*n+1)*(1 - x^n)^(n+1)*A(x)^n + ...
%e A357152 also
%e A357152 -A(x)^5 = ... + x^(-3)*(A(x) - x^(-2))^(-1)*A(x)^2 + x^(-1)*A(x) + x*(A(x) - 1) + x^3*(A(x) - x)^2/A(x) + x^5*(1 - x^2)^3/A(x)^2 + x^7*(A(x) - x^3)^4/A(x)^3 + ... + x^(2*n+1)*(A(x) - x^n)^(n+1)/A(x)^n + ...
%o A357152 (PARI) {a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0);
%o A357152 A[#A] = polcoeff(Ser(A)^2 - sum(n=-#A\2-1, #A\2+1, x^(2*n+1) * (1 - x^n +x*O(x^#A))^(n+1) * Ser(A)^n  ), #A-2); ); A[n+1]}
%o A357152 for(n=0, 30, print1(a(n), ", "))
%Y A357152 Cf. A356783, A357151, A357153, A357154, A357155.
%K A357152 nonn
%O A357152 0,3
%A A357152 _Paul D. Hanna_, Sep 16 2022