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A270121 Denominators in a perturbed Engel series.

%I A270121 #20 Nov 20 2020 06:52:23
%S A270121 7,112,403200,1755760043520000,
%T A270121 53695136666462381094317154204367872000000
%N A270121 Denominators in a perturbed Engel series.
%C A270121 The sum of the series 6/a(1)+1/a(2)+1/a(3)+... is a transcendental number, and has a continued fraction expansion whose coefficients are given explicitly in terms of the sequence a(n) and the ratios a(n+1)/a(n).
%H A270121 Amiram Eldar, <a href="/A270121/b270121.txt">Table of n, a(n) for n = 1..8</a>
%H A270121 Andrew N. W. Hone, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Hone/hone3.html">Curious continued fractions, nonlinear recurrences and transcendental numbers</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.8.4.
%H A270121 Andrew N. W. Hone, <a href="http://arxiv.org/abs/1509.05019">Continued fractions for some transcendental numbers</a>, arXiv:1509.05019 [math.NT], 2015-2016, Monatsh. Math. DOI: 10.1007/s00605-015-0844-2.
%F A270121 The sequence is generated by taking a(n+1)=b(n-1)*a(n)*(1+n*a(n)), b(n)=a(n+1)/a(n) for n>=1 with initial values a(1)=7,b(0)=2. Alternatively, if a(1)=7,a(2)=112 are given then a(n+1)*a(n-1)=a(n)^2*(1+n*a(n)) for n>=2.
%F A270121 Sum_{n>=1} 1/a(n) = -5/7 + A270137. - _Amiram Eldar_, Nov 20 2020
%t A270121 a[1] = 7; a[2] = 112;
%t A270121 a[n_] := a[n] = (a[n-1]^2 (1+(n-1)a[n-1]))/a[n-2];
%t A270121 Array[a, 5] (* _Jean-François Alcover_, Dec 16 2018 *)
%Y A270121 Cf. A112373, A114552, A114550, A270137.
%K A270121 nonn
%O A270121 1,1
%A A270121 _Andrew Hone_, Mar 11 2016