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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
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Here AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) denotes the arithmetic-geometric mean. Limit a(n+1)/a(n) = 3. LINKS Paul D. Hanna, Table of n, a(n) for n = 0..1000 FORMULA 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 MATHEMATICA 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 *) PROG (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), ", ")) CROSSREFS Cf. A245931. Sequence in context: A273359 A251422 A018836 * A274323 A297740 A297741 Adjacent sequences:  A245929 A245930 A245931 * A245933 A245934 A245935 KEYWORD nonn AUTHOR Paul D. Hanna, Aug 14 2014 STATUS approved

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Last modified June 22 10:52 EDT 2021. Contains 345375 sequences. (Running on oeis4.)