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A380068 G.f. A(x) satisfies 1/2 = Sum_{n=-oo..+oo} x^n * (A(x) + x^n)^(2*n-1).

%I A380068 #8 Jan 23 2025 11:52:52
%S A380068 1,4,36,312,3440,40956,518160,6806320,92021528,1271748364,17886165344,
%T A380068 255159368504,3683262020928,53700117957756,789606760314200,
%U A380068 11696040806690484,174362944317804916,2614112736300210308,39388817610142696848,596167096482669128248,9059675614901834999980,138177866602598729509112
%N A380068 G.f. A(x) satisfies 1/2 = Sum_{n=-oo..+oo} x^n * (A(x) + x^n)^(2*n-1).
%H A380068 Paul D. Hanna, <a href="/A380068/b380068.txt">Table of n, a(n) for n = 0..300</a>
%F A380068 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A380068 (1) 1/2 = Sum_{n=-oo..+oo} x^n * (A(x) + x^n)^(2*n-1).
%F A380068 (2) 1/2 = Sum_{n=-oo..+oo} x^(2*n^2) / (1 + x^n*A(x))^(2*n+1).
%F A380068 a(n) ~ c * d^n / n^(3/2), where d = 16.39184422422602948598371145634761000079... and c = 0.4231059327883494572901551013193022... - _Vaclav Kotesovec_, Jan 23 2025
%e A380068 G.f.: A(x) = 1 + 4*x + 36*x^2 + 312*x^3 + 3440*x^4 + 40956*x^5 + 518160*x^6 + 6806320*x^7 + 92021528*x^8 + 1271748364*x^9 + 17886165344*x^10 + ...
%e A380068 SPECIFIC VALUES.
%e A380068 A(t) = 9/5 at t = 0.060810040367940244892240134748250077713967840944862...
%e A380068 A(t) = 7/4 at t = 0.060471316741109796362890999165339625209169541570118...
%e A380068 A(t) = 5/3 at t = 0.059455864011187363622702920671351845740910151873822...
%e A380068 A(t) = 3/2 at t = 0.055175405333610355588278758628977431335363340728879...
%e A380068 A(t) = 4/3 at t = 0.046413573549935696160990703887268428961721362286826...
%e A380068 A(t) = 5/4 at t = 0.039506911682991228951042053988737197176348217600170...
%e A380068 A(1/17) = 1.6311797797272774131428286483151621703548116404225...
%e A380068 A(1/18) = 1.5105106929462926658533664976702138681313516080377...
%e A380068 A(1/20) = 1.3892813212708752419341068817323469222865996600644...
%e A380068 A(1/25) = 1.2551873360504999226413532789756472196341294007207...
%o A380068 (PARI) {a(n) = my(V=[1]); for(i=1,n, V = concat(V,0); A = Ser(V);
%o A380068 V[#V] = polcoef(-2 + 4*sum(n=-#V,#V, x^n * (A + x^n)^(2*n-1) ),#V-1) ); V[n+1]}
%o A380068 for(n=0,30,print1(a(n),", "))
%Y A380068 Cf. A379763, A379765.
%K A380068 nonn
%O A380068 0,2
%A A380068 _Paul D. Hanna_, Jan 23 2025