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A357151 Coefficients in the power series A(x) such that: A(x) = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.

%I A357151 #12 Sep 16 2022 22:01:44
%S A357151 1,1,3,13,60,299,1586,8697,49117,283437,1664128,9908903,59694494,
%T A357151 363179981,2228272706,13771458148,85655772108,535759514193,
%U A357151 3367801361510,21264574306632,134804893426581,857682458939905,5474890014327326,35053167752718368,225046818744827456
%N A357151 Coefficients in the power series A(x) such that: A(x) = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.
%C A357151 Related identity: 0 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1).
%H A357151 Paul D. Hanna, <a href="/A357151/b357151.txt">Table of n, a(n) for n = 0..400</a>
%F A357151 G.f. A(x) = Sum_{n>=0} a(n) * x^n satisfies the following relations.
%F A357151 (1) A(x) = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.
%F A357151 (2) x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 - x^(n+1))^n * A(x)^n ).
%F A357151 (3) -x*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (1 - x^(n+1)*A(x))^n.
%F A357151 (4) -A(x)^4 = Sum_{n=-oo..+oo} x^(2*n+1) * (A(x) - x^n)^(n+1) / A(x)^n.
%F A357151 (5) 0 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n*A(x))^(n+1) / A(x)^n.
%F A357151 (6) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (A(x) - x^(n+1))^n.
%e A357151 G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 60*x^4 + 299*x^5 + 1586*x^6 + 8697*x^7 + 49117*x^8 + 283437*x^9 + 1664128*x^10 + 9908903*x^11 + 59694494*x^12 + ...
%e A357151 such that
%e A357151 A(x) = ... + 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 A357151 also
%e A357151 -A(x)^4 = ... + 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 A357151 (PARI) {a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0);
%o A357151 A[#A] = polcoeff(Ser(A) - 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 A357151 for(n=0, 30, print1(a(n), ", "))
%Y A357151 Cf. A356783, A357152, A357153, A357154, A357155.
%K A357151 nonn
%O A357151 0,3
%A A357151 _Paul D. Hanna_, Sep 16 2022