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A380677 G.f. A(x) satisfies 1 = Sum_{n=-oo..+oo} x^(2*n) * (x^n - A(x))^(3*n+1).

%I A380677 #11 Feb 03 2025 02:22:31
%S A380677 1,2,8,36,198,1128,6837,42690,273960,1792650,11922735,80342746,
%T A380677 547403208,3764568202,26097746670,182183863242,1279566641040,
%U A380677 9035527984360,64109825254786,456834687004440,3267926616628182,23458797921291994,168936073477132102,1220121029135864026,8835737467337361482
%N A380677 G.f. A(x) satisfies 1 = Sum_{n=-oo..+oo} x^(2*n) * (x^n - A(x))^(3*n+1).
%H A380677 Paul D. Hanna, <a href="/A380677/b380677.txt">Table of n, a(n) for n = 0..350</a>
%F A380677 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A380677 (1) 1 = Sum_{n=-oo..+oo} x^(2*n) * (x^n - A(x))^(3*n+1).
%F A380677 (2) 1 = Sum_{n=-oo..+oo} x^(3*n*(n-1)) / (1 - x^n*A(x))^(3*n-1).
%e A380677 G.f.: A(x) = 1 + 2*x + 8*x^2 + 36*x^3 + 198*x^4 + 1128*x^5 + 6837*x^6 + 42690*x^7 + 273960*x^8 + 1792650*x^9 + 11922735*x^10 + ...
%e A380677 SPECIFIC VALUES.
%e A380677 A(t) = 7/4 at t = 0.12654949614445186746403892264694555335923498557738...
%e A380677   where 1 = Sum_{n=-oo..+oo} t^(2*n) * (t^n - 7/4)^(3*n+1).
%e A380677 A(t) = 5/3 at t = 0.12374694612565134762563311753154796236873902596812...
%e A380677 A(t) = 3/2 at t = 0.11392195456863186572686610752037791827642247932473...
%e A380677 A(t) = 4/3 at t = 0.09535917714046949923896929084305426642940930464927...
%e A380677 A(t) = 5/4 at t = 0.08098320583796566321668508295130093344916245020730...
%e A380677 A(1/8) = 1.69987163237671043867918157348979527169465395859405...
%e A380677   where 1 = Sum_{n=-oo..+oo} (1/8)^(2*n) * ((1/8)^n - A(1/8))^(3*n+1).
%e A380677 A(1/9) = 1.46724009425513930419976858432180568713155056224164...
%e A380677 A(1/10) = 1.3665270076239843695076027726524469708778850053524...
%e A380677 A(1/11) = 1.3048130783240200786482939740924774873262324649207...
%e A380677 A(1/12) = 1.2620494023042372384830602119971826992309809007730...
%e A380677 A(1/14) = 1.2057100150678855865365454675611764497376238367914...
%e A380677 A(1/16) = 1.1698113057379453133949062841882391284824341375308...
%o A380677 (PARI) {a(n) = my(V=[1]); for(i=1, n, V = concat(V, 0); A = Ser(V);
%o A380677 V[#V] = polcoef(-1 + sum(n=-#V, #V, x^(2*n) * (x^n - A)^(3*n+1) ), #V-1) ); H=A; V[n+1]}
%o A380677 for(n=0, 30, print1(a(n), ", "))
%Y A380677 Cf. A380067, A355866, A380676.
%K A380677 nonn
%O A380677 0,2
%A A380677 _Paul D. Hanna_, Feb 02 2025