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A357402 Coefficients in the power series A(x) such that: 2 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.

%I A357402 #8 Oct 08 2022 00:26:48
%S A357402 1,2,8,42,236,1420,8976,58644,393200,2689522,18694164,131658910,
%T A357402 937490780,6737990172,48816739048,356142597586,2614103310384,
%U A357402 19291118713324,143044431901580,1065237986700788,7963426677825000,59741019702076168,449601401992383464,3393484429948103486
%N A357402 Coefficients in the power series A(x) such that: 2 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.
%C A357402 Related identity: 0 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1).
%C A357402 a(n) = Sum_{k=0..n} A357400(n,k) * 2^k, for n >= 0.
%H A357402 Paul D. Hanna, <a href="/A357402/b357402.txt">Table of n, a(n) for n = 0..300</a>
%F A357402 G.f. A(x) = Sum_{n>=0} a(n) * x^n satisfies the following relations.
%F A357402 (1) 2 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.
%F A357402 (2) 2*x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 - x^(n+1))^n * A(x)^n ).
%F A357402 (3) -2*x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (1 - x^(n+1)*A(x))^n.
%F A357402 (4) -2*A(x)^3 = Sum_{n=-oo..+oo} x^(2*n+1) * (A(x) - x^n)^(n+1) / A(x)^n.
%F A357402 (5) 0 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n*A(x))^(n+1) / A(x)^n.
%F A357402 (6) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (A(x) - x^(n+1))^n.
%e A357402 G.f.: A(x) = 1 + 2*x + 8*x^2 + 42*x^3 + 236*x^4 + 1420*x^5 + 8976*x^6 + 58644*x^7 + 393200*x^8 + 2689522*x^9 + 18694164*x^10 + ...
%e A357402 such that
%e A357402 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 A357402 also
%e A357402 -2*A(x)^3 = ... + 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 A357402 (PARI) {a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0);
%o A357402 A[#A] = polcoeff(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 A357402 for(n=0, 30, print1(a(n), ", "))
%Y A357402 Cf. A357400, A356783, A357403, A357404, A357405.
%K A357402 nonn
%O A357402 0,2
%A A357402 _Paul D. Hanna_, Sep 26 2022