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A380676 G.f. A(x) satisfies 2 = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) + x^n)^(3*n+1).

%I A380676 #10 Feb 03 2025 02:22:26
%S A380676 1,2,9,76,591,5127,46919,444617,4333010,43132310,436715297,4483520704,
%T A380676 46564078707,488335074439,5164287656762,55010054836724,
%U A380676 589682412920880,6356441723399838,68858811108713642,749250723117079260,8185098919015604558,89739660783143322586,987110817010576637569
%N A380676 G.f. A(x) satisfies 2 = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) + x^n)^(3*n+1).
%H A380676 Paul D. Hanna, <a href="/A380676/b380676.txt">Table of n, a(n) for n = 0..350</a>
%F A380676 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A380676 (1) 2 = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) + x^n)^(3*n+1).
%F A380676 (2) 2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(3*n-2)) / (1 + x^n*A(x))^(3*n-1).
%e A380676 G.f.: A(x) = 1 + 2*x + 9*x^2 + 76*x^3 + 591*x^4 + 5127*x^5 + 46919*x^6 + 444617*x^7 + 4333010*x^8 + 43132310*x^9 + 436715297*x^10 + ...
%e A380676 SPECIFIC VALUES.
%e A380676 A(t) = 3/2 at t = 0.084454810721317538501174440773777047952092460562060...
%e A380676   where 2 = Sum_{n=-oo..+oo} (-t)^n * (3/2 + t^n)^(3*n+1).
%e A380676 A(t) = 4/3 at t = 0.077952215522932621280995556726745992779521168178442...
%e A380676 A(t) = 5/4 at t = 0.069865542488187377549700484712724108090103217291400...
%e A380676 A(t) = 6/5 at t = 0.062525019563729453209334340397151869258204650105887...
%e A380676 A(1/12) = 1.4451475449531942766582635648883506035661276873944...
%e A380676   where 2 = Sum_{n=-oo..+oo} (-1/12)^n * (A(1/12) + (1/12)^n)^(3*n+1).
%e A380676 A(1/13) = 1.3197666375699291221191258833369709715040515804644...
%e A380676 A(1/14) = 1.2629677124586701325494126247872966004241466655536...
%e A380676 A(1/15) = 1.2263276036037963341062042248250428743844880153971...
%e A380676 A(1/16) = 1.1998529038743458677434930677034050910039899372219...
%o A380676 (PARI) {a(n) = my(V=[1]); for(i=1, n, V = concat(V, 0); A = Ser(V);
%o A380676 V[#V] = polcoef(2 - sum(n=-#V, #V, (-1)^n * x^n * (A + x^n)^(3*n+1) ), #V-1) ); H=A; V[n+1]}
%o A380676 for(n=0, 30, print1(a(n), ", "))
%Y A380676 Cf. A380067, A355866, A380677.
%K A380676 nonn
%O A380676 0,2
%A A380676 _Paul D. Hanna_, Feb 02 2025