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A245931 G.f.: 1 / sqrt( AGM((1 - 3*x)^2, (1 + x)^2) ).

%I A245931 #22 Sep 29 2019 11:55:14
%S A245931 1,1,1,1,3,11,31,71,157,397,1141,3301,9079,24207,65339,182131,517307,
%T A245931 1467067,4128859,11606683,32835433,93588097,267745149,766331573,
%U A245931 2193690811,6289737611,18081071971,52099389811,150344751721,434277817873,1255708046341,3635404626381,10538677900781,30585912440557
%N A245931 G.f.: 1 / sqrt( AGM((1 - 3*x)^2, (1 + x)^2) ).
%C A245931 Here AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) denotes the arithmetic-geometric mean.
%C A245931 Self-convolution yields A245930.
%C A245931 Limit a(n+1)/a(n) = 3.
%H A245931 Paul D. Hanna, <a href="/A245931/b245931.txt">Table of n, a(n) for n = 0..1000</a>
%F A245931 G.f.: 1 / sqrt( AGM((1-x)^2 + 4*x^2, (1-x)^2 - 4*x^2) ).
%F A245931 G.f.: 1 / sqrt( AGM((1-x)^2, sqrt((1-x)^4 - 16*x^4)) ).
%F A245931 a(n) ~ 3^(n+1) / (4*n*sqrt(Pi*log(n))) * (1 + (log(3) - 3*log(2) - gamma) / (2*log(n)) + (3*gamma^2/8 + 9*gamma*log(2)/4 - 3*gamma*log(3)/4 + 27*log(2)^2/8 - 9*log(3)*log(2)/4 + 3*log(3)^2/8 - Pi^2/16) / log(n)^2), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Sep 29 2019
%e A245931 G.f.: A(x) = 1 + x + x^2 + x^3 + 3*x^4 + 11*x^5 + 31*x^6 + 71*x^7 + 157*x^8 +...
%e A245931 where
%e A245931 1/A(x) = 1 - x - 2*x^4 - 6*x^5 - 12*x^6 - 20*x^7 - 42*x^8 - 126*x^9 - 392*x^10 - 1080*x^11 - 2722*x^12 - 6886*x^13 - 18612*x^14 - 52780*x^15 - 149564*x^16 -...
%e A245931 equals sqrt( AGM((1 - 3*x)^2, (1 + x)^2) ).
%e A245931 The logarithmic derivative of the g.f. begins:
%e A245931 A'(x)/A(x) = 1 + x + x^2 + 9*x^3 + 41*x^4 + 121*x^5 + 281*x^6 + 673*x^7 + 2017*x^8 + 6721*x^9 + 21121*x^10 + 61065*x^11 + 171497*x^12 +...+ A245932(n)*x^n +...
%t A245931 Simplify[CoefficientList[Series[Sqrt[(2*EllipticK[1 - (1 - 3*x)^4/(1 + x)^4]) / Pi] / (1 + x), {x, 0, 30}], x]] (* _Vaclav Kotesovec_, Sep 27 2019 *)
%o A245931 (PARI) {a(n)=local(A=1); A = 1 / sqrt( agm((1-3*x)^2, (1+x)^2 +x*O(x^n)) ); polcoeff(A,n)}
%o A245931 for(n=0,35,print1(a(n),", "))
%Y A245931 Cf. A245930, A245932 (log), A227846.
%K A245931 nonn
%O A245931 0,5
%A A245931 _Paul D. Hanna_, Aug 14 2014