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A245932
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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.
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3
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1, 1, 1, 9, 41, 121, 281, 673, 2017, 6721, 21121, 61065, 171497, 495769, 1488761, 4509793, 13468897, 39688609, 116869153, 346788009, 1035199817, 3090560089, 9200749433, 27347417281, 81352371841, 242426988961, 723125351521, 2156829477609, 6430792717001, 19174372701241, 57194628447641
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OFFSET
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0,4
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COMMENTS
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Here AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) denotes the arithmetic-geometric mean.
Limit a(n+1)/a(n) = 3.
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LINKS
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FORMULA
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G.f.: A(x) = 1 + x + x^2 + 9*x^3 + 41*x^4 + 121*x^5 + 281*x^6 + 673*x^7 +...
As a logarithmic expansion,
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 +...
where
exp(L(x)) = 1 + x + x^2 + x^3 + 3*x^4 + 11*x^5 + 31*x^6 + 71*x^7 + 157*x^8 +...
equals 1 / sqrt( AGM((1 - 3*x)^2, (1 + x)^2) ).
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
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MATHEMATICA
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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 *)
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PROG
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(PARI) /* As the logarithmic derivative of A245931: */
{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)}
for(n=0, 35, print1(a(n), ", "))
(PARI) /* As the logarithm of g.f. of A245931 (offset = 1): */
{a(n)=local(A=1); A = -log( agm((1-3*x)^2, (1+x)^2 +x*O(x^n)) )/2; n*polcoeff(A, n)}
for(n=1, 35, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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