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%I #29 Sep 27 2019 11:58:36
%S 1,2,6,20,66,212,756,3320,11346,-11068,14556,7202120,18928476,
%T -1376971048,-3526491144,394396083920,1016723438706,-148493230507228,
%U -383613651929844,71479338751223720,184867683069498036
%N E.g.f.: AGM(1, exp(4x)), where AGM(x, y) is the arithmetic-geometric mean of Gauss.
%C Conjecture: limit |a(n)/n!|^(-1/n) = r exists and is finite with r<0.8...
%C What is the radius of convergence of the e.g.f. as a power series in x?
%C r = Pi/4. - _Vaclav Kotesovec_, Sep 27 2019
%H Vaclav Kotesovec, <a href="/A174846/b174846.txt">Table of n, a(n) for n = 0..430</a>
%F E.g.f.: exp(2x)*AGM(1, cosh(2x)).
%F E.g.f.: exp(2x)*AGM( cosh(x)^2, sqrt(cosh(2x)) ).
%e E.g.f.: A(x) = 1 + 2*x + 6*x^2/2! + 20*x^3/3! + 66*x^4/4! +...
%e Special value:
%e A(log(2)/8) = Pi^(3/2)*sqrt(8)/gamma(1/4)^2 = 1.19814023473...
%t nmax = 20; CoefficientList[Series[E^(4*x)*Pi / (2*EllipticK[1 - E^(-8*x)]), {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Sep 27 2019 *)
%o (PARI) {a(n)=n!*polcoeff(agm(1,exp(4*x+x*O(x^n))),n)}
%K sign
%O 0,2
%A _Paul D. Hanna_, Jan 24 2011