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A379765 G.f. A(x) satisfies 1/2 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (A(x) + (-x)^n)^(2*n-1).

%I A379765 #7 Jan 22 2025 03:27:16
%S A379765 1,4,16,108,764,5772,45608,372112,3110868,26511720,229465696,
%T A379765 2011560120,17823251908,159361875452,1436070211128,13029220181024,
%U A379765 118919107720504,1091130632899108,10058749510188900,93119868866604632,865350260237277984,8069341311245971172,75482617925071807900
%N A379765 G.f. A(x) satisfies 1/2 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (A(x) + (-x)^n)^(2*n-1).
%C A379765 Related identity: 0 = Sum_{n=-oo..+oo} x^(2*n) * (y - x^n)^(n+1), which holds for all y as a formal power series in x.
%H A379765 Paul D. Hanna, <a href="/A379765/b379765.txt">Table of n, a(n) for n = 0..300</a>
%F A379765 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A379765 (1) 1/2 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (A(x) + (-x)^n)^(2*n-1).
%F A379765 (2) 1/2 = Sum_{n=-oo..+oo} x^(n*(2*n-1)) / (1 + (-x)^n*A(x))^(2*n+1).
%F A379765 a(n) ~ c * d^n / n^(3/2), where d = 10.0028789085783346757183844487387995725278... and c = 0.82085228741035116338050876364461533993... - _Vaclav Kotesovec_, Jan 22 2025
%e A379765 G.f.: A(x) = 1 + 4*x + 16*x^2 + 108*x^3 + 764*x^4 + 5772*x^5 + 45608*x^6 + 372112*x^7 + 3110868*x^8 + 26511720*x^9 + 229465696*x^10 + ...
%e A379765 SPECIFIC VALUES.
%e A379765 A(t) = 2 at t = 0.09653361905915170411984272932017391451579223633328063596...
%e A379765 A(t) = 3/2 at t = 0.075349068495101237164879111573650564301786145264462860...
%e A379765 A(t) = 4/3 at t = 0.059270539047675430011940298576693796977637316335946787...
%e A379765 A(t) = 5/4 at t = 0.048395689470367112406846758430274833454550275773025303...
%e A379765 A(1/11) = 1.7859531236503504891314901027866679467841212478816...
%e A379765 A(1/12) = 1.6197331904782587957364168880788048296765610762165...
%e A379765 A(1/13) = 1.5208322951137414977098221192098933277226208754204...
%e A379765 A(1/14) = 1.4524825965114467294893588185221903177323240271997...
%e A379765 A(1/20) = 1.2612955244894656446063844025636148211057476645966...
%o A379765 (PARI) {a(n) = my(V=[1]); for(i=1,n, V=concat(V,0); A = Ser(V);
%o A379765 V[#V] = polcoef(-2 + 4*sum(n=-#V,#V, (-1)^n * x^(2*n) * (A + (-x)^n)^(2*n-1) ),#V-1) );V[n+1]}
%o A379765 for(n=0,30,print1(a(n),", "))
%Y A379765 Cf. A379763.
%K A379765 nonn
%O A379765 0,2
%A A379765 _Paul D. Hanna_, Jan 22 2025