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A245932 G.f.: G'(x) / G(x) where G(x) = 1 / sqrt( AGM((1 - 3*x)^2, (1 + x)^2) ) is the g.f. of A245931. 3

%I

%S 1,1,1,9,41,121,281,673,2017,6721,21121,61065,171497,495769,1488761,

%T 4509793,13468897,39688609,116869153,346788009,1035199817,3090560089,

%U 9200749433,27347417281,81352371841,242426988961,723125351521,2156829477609,6430792717001,19174372701241,57194628447641

%N G.f.: G'(x) / G(x) where G(x) = 1 / sqrt( AGM((1 - 3*x)^2, (1 + x)^2) ) is the g.f. of A245931.

%C Here AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) denotes the arithmetic-geometric mean.

%C Limit a(n+1)/a(n) = 3.

%H Paul D. Hanna, <a href="/A245932/b245932.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: A(x) = 1 + x + x^2 + 9*x^3 + 41*x^4 + 121*x^5 + 281*x^6 + 673*x^7 +...

%F As a logarithmic expansion,

%F L(x) = x + x^2/2 + x^3/3 + 9*x^4/4 + 41*x^5/5 + 121*x^6/6 + 281*x^7/7 + 673*x^8/8 + 2017*x^9/9 + 6721*x^10/10 +...

%F where

%F exp(L(x)) = 1 + x + x^2 + x^3 + 3*x^4 + 11*x^5 + 31*x^6 + 71*x^7 + 157*x^8 +...

%F equals 1 / sqrt( AGM((1 - 3*x)^2, (1 + x)^2) ).

%F a(n) ~ 3^(n+1) / (2*log(n)) * (1 + (log(3) - 3*log(2) - gamma) / log(n)), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Sep 30 2019

%t Simplify[CoefficientList[Series[D[Log[Sqrt[(2*EllipticK[1 - (1 - 3*x)^4/(1 + x)^4])/Pi] / (1 + x)], x], {x, 0, 30}], x]] (* _Vaclav Kotesovec_, Sep 27 2019 *)

%o (PARI) /* As the logarithmic derivative of A245931: */

%o {a(n)=local(G=1); G = 1 / sqrt( agm((1-3*x)^2, (1+x)^2 +x^2*O(x^n)) ); polcoeff(G'/G,n)}

%o for(n=0,35,print1(a(n),", "))

%o (PARI) /* As the logarithm of g.f. of A245931 (offset = 1): */

%o {a(n)=local(A=1); A = -log( agm((1-3*x)^2, (1+x)^2 +x*O(x^n)) )/2; n*polcoeff(A,n)}

%o for(n=1,35,print1(a(n),", "))

%Y Cf. A245931.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Aug 14 2014

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Last modified August 5 22:07 EDT 2020. Contains 336214 sequences. (Running on oeis4.)