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A382323 G.f. A(x) satisfies -3 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (1 + x^n)^(n+1) * A(x)^n.

%I A382323 #13 Mar 22 2025 18:50:20
%S A382323 1,3,18,150,1323,12486,123069,1253595,13089576,139367370,1507353966,
%T A382323 16515098985,182913374493,2044565139303,23035036108755,
%U A382323 261312501113193,2982280058702499,34217698991867058,394470188685557271,4566935001939261414,53076293916648500439,618991948535588040078
%N A382323 G.f. A(x) satisfies -3 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (1 + x^n)^(n+1) * A(x)^n.
%C A382323 Related identity: 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (1 + x^n)^(n+1).
%C A382323 Formulas (5) and (6) hold true for all nonzero A(x).
%H A382323 Paul D. Hanna, <a href="/A382323/b382323.txt">Table of n, a(n) for n = 0..301</a>
%F A382323 G.f. A(x) = Sum_{n>=0} a(n) * x^n satisfies the following formulas.
%F A382323 (1) -3 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (1 + x^n)^(n+1) * A(x)^n.
%F A382323 (2) 3*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (A(x) + x^n)^(n+1) / A(x)^n.
%F A382323 (3) -3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-3)+1) / ( (1 + x^n)^(n-1) * A(x)^n ).
%F A382323 (4) 3*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-3)+1) * A(x)^n / (1 + x^n*A(x))^(n-1).
%F A382323 (5) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (1 + x^n*A(x))^(n+1) / A(x)^n.
%F A382323 (6) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-3)+1) * A(x)^n / (A(x) + x^n)^(n-1).
%e A382323 G.f.: A(x) = 1 + 3*x + 18*x^2 + 150*x^3 + 1323*x^4 + 12486*x^5 + 123069*x^6 + 1253595*x^7 + 13089576*x^8 + 139367370*x^9 + ...
%o A382323 (PARI) {a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0);
%o A382323 A[#A] = polcoef(3 + sum(m=-#A, #A, (-1)^m * x^(2*m+1) * (1 + x^m +x*O(x^#A))^(m+1) * Ser(A)^m  ), #A-2); ); A[n+1]}
%o A382323 for(n=0, 30, print1(a(n), ", "))
%Y A382323 Cf. A356783, A380557, A382322.
%K A382323 nonn
%O A382323 0,2
%A A382323 _Paul D. Hanna_, Mar 21 2025