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A357161 Coefficients in the power series A(x) such that: A(x) = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1) * A(x)^n.

%I A357161 #10 Sep 20 2022 21:19:58
%S A357161 1,1,3,15,71,378,2087,12006,70910,428021,2627731,16358961,103027423,
%T A357161 655236314,4202210514,27145925685,176474644608,1153679423108,
%U A357161 7579526316199,50017854059557,331390828183765,2203548061830875,14700363755114949,98363233394747546
%N A357161 Coefficients in the power series A(x) such that: A(x) = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1) * A(x)^n.
%C A357161 Compare to A357151.
%C A357161 Related identity: 0 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1).
%C A357161 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 A357161 Paul D. Hanna, <a href="/A357161/b357161.txt">Table of n, a(n) for n = 0..400</a>
%F A357161 G.f. A(x) = Sum_{n>=0} a(n) * x^n satisfies the following relations.
%F A357161 (1) A(x) = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1) * A(x)^n.
%F A357161 (2) x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 - x^(n+2))^n * A(x)^n ).
%F A357161 (3) -x*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (1 - x^(n+2)*A(x))^n.
%F A357161 (4) -A(x)^4 = Sum_{n=-oo..+oo} x^(3*n+2) * (A(x) - x^(n-1))^(n+1) / A(x)^n.
%F A357161 (5) 0 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1)*A(x))^(n+1) / A(x)^n.
%F A357161 (6) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (A(x) - x^(n+2))^n.
%e A357161 G.f.: A(x) = 1 + x + 3*x^2 + 15*x^3 + 71*x^4 + 378*x^5 + 2087*x^6 + 12006*x^7 + 70910*x^8 + 428021*x^9 + 2627731*x^10 + ...
%e A357161 such that
%e A357161 A(x) = ... + 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 A357161 also
%e A357161 -A(x)^4 = ... + 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 A357161 (PARI) {a(n) = my(A=[1]); for(i=0,n, A = concat(A,0);
%o A357161 A[#A] = polcoeff(Ser(A) - 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 A357161 for(n=0,30, print1(a(n),", "))
%Y A357161 Cf. A357151, A357160, A357162, A357163, A357164, A357165.
%K A357161 nonn
%O A357161 0,3
%A A357161 _Paul D. Hanna_, Sep 17 2022