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A357160 Coefficients in the power series A(x) such that: 1 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1) * A(x)^n.

%I A357160 #9 Sep 20 2022 19:21:40
%S A357160 1,1,2,8,24,88,313,1187,4549,17898,71324,288365,1177729,4856051,
%T A357160 20178061,84427850,355375253,1503849591,6394015744,27301536104,
%U A357160 117020066991,503313598572,2171633107742,9396938664272,40769489510945,177313714453588,772906669281227,3376119803594888
%N A357160 Coefficients in the power series A(x) such that: 1 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1) * A(x)^n.
%C A357160 Compare to A356783.
%C A357160 Related identity: 0 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1).
%C A357160 Related identity: 0 = Sum_{n=-oo..+oo} x^(k*n) * (y - x^(n+1-k))^n, which holds for any positive integer k and real y.
%H A357160 Paul D. Hanna, <a href="/A357160/b357160.txt">Table of n, a(n) for n = 0..400</a>
%F A357160 G.f. A(x) = Sum_{n>=0} a(n) * x^n satisfies the following relations.
%F A357160 (1) 1 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1) * A(x)^n.
%F A357160 (2) x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 - x^(n+2))^n * A(x)^n ).
%F A357160 (3) -x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (1 - x^(n+2)*A(x))^n.
%F A357160 (4) -A(x)^3 = Sum_{n=-oo..+oo} x^(3*n+2) * (A(x) - x^(n-1))^(n+1) / A(x)^n.
%F A357160 (5) 0 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1)*A(x))^(n+1) / A(x)^n.
%F A357160 (6) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (A(x) - x^(n+2))^n.
%e A357160 G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 24*x^4 + 88*x^5 + 313*x^6 + 1187*x^7 + 4549*x^8 + 17898*x^9 + 71324*x^10 + ...
%e A357160 such that
%e A357160 1 = ... + x^(-4)*(1 - 1/x^3)^(-1)/A(x)^2 + x^(-1)/A(x) + x^2*(1 - 1/x) + x^5*0*A(x) + x^8*(1 - x)^3*A(x)^2 + x^11*(1 - x^2)^4*A(x)^3 + ... + x^(3*n+2)*(1 - x^(n-1))^(n+1)*A(x)^n + ...
%e A357160 also
%e A357160 -A(x)^3 = ... + x^(-4)*(A(x) - 1/x^3)^(-1)*A(x)^2 + x^(-1)*A(x) + x^2*(A(x) - 1/x) + x^5*(A(x) - 1)^2/A(x) + x^8*(A(x) - x)^3/A(x)^2 + x^11*(A(x) - x^2)^4/A(x)^3 + ... + x^(3*n+2)*(A(x) - x^(n-1))^(n+1)/A(x)^n + ...
%o A357160 (PARI) {a(n) = my(A=[1]); for(i=0,n, A = concat(A,0);
%o A357160 A[#A] = polcoeff(1 - sum(n=-#A\3-2,#A\3+2, x^(3*n+2) * (1 - x^(n-1) +x*O(x^#A))^(n+1) * Ser(A)^n  ),#A-2); );A[n+1]}
%o A357160 for(n=0,30, print1(a(n),", "))
%Y A357160 Cf. A356783, A357161, A357162, A357163, A357164, A357165.
%K A357160 nonn
%O A357160 0,3
%A A357160 _Paul D. Hanna_, Sep 17 2022