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A208034 G.f.: exp( Sum_{n>=1} 2*Pell(n)^2 * x^n/n ), where Pell(n) = A000129(n).

%I A208034 #28 Oct 31 2024 10:27:26
%S A208034 1,2,6,26,122,602,3062,15906,83906,447842,2412566,13094490,71513210,
%T A208034 392592410,2164815590,11982792386,66548673282,370672213826,
%U A208034 2069974290726,11586244722202,64986102400122,365183031749722,2055594717395926,11588727763937506,65425688924696002
%N A208034 G.f.: exp( Sum_{n>=1} 2*Pell(n)^2 * x^n/n ), where Pell(n) = A000129(n).
%C A208034 Conjectures: For all positive integers k,
%C A208034 (1) exp( Sum_{n>=1} 2*Pell(n)^(2*k) * x^n/n ) is an integer series;
%C A208034 (2) exp( Sum_{n>=1} 2*Pell(n)^(2*k-1) * x^n/n ) is NOT an integer series;
%C A208034 (3) exp( Sum_{n>=1} Pell(n)^(2*k) * x^n/n ) is NOT an integer series.
%H A208034 Andrew Howroyd, <a href="/A208034/b208034.txt">Table of n, a(n) for n = 0..200</a>
%F A208034 G.f.: sqrt(1 + x) / (1 - 6*x + x^2)^(1/4).
%F A208034 a(n) ~ 2^(1/8) * (1 + sqrt(2))^(2*n) / (Gamma(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Oct 31 2024
%e A208034 G.f.: A(x) = 1 + 2*x + 6*x^2 + 26*x^3 + 122*x^4 + 602*x^5 + 3062*x^6 + ...
%e A208034 such that, by definition,
%e A208034 log(A(x))/2 = x + 2^2*x^2/2 + 5^2*x^3/3 + 12^2*x^4/4 + 29^2*x^5/5 + 70^2*x^6/6 + 169^2*x^7/7 + 408^2*x^8/8 + ... + Pell(n)^2*x^n/n + ...
%t A208034 CoefficientList[Series[Sqrt[1 + x] / (1 - 6 x + x^2)^(1/4), {x, 0, 33}], x] (* _Vincenzo Librandi_, Feb 26 2018 *)
%o A208034 (PARI) {Pell(n)=polcoeff(x/(1-2*x-x^2 +x*O(x^n)),n)}
%o A208034 {a(n)=polcoeff(exp(sum(m=1,n,2*Pell(m)^2*x^m/m) +x*O(x^n)),n)}
%o A208034 for(n=0,30,print1(a(n),", "))
%o A208034 (PARI) seq(n)={Vec(sqrt(1 + x + O(x^n)) / sqrt(sqrt(1 - 6*x + x^2 + O(x^n))))} \\ _Andrew Howroyd_, Feb 25 2018
%Y A208034 Cf. A000129, A208055, A208056, A204061, A204062.
%K A208034 nonn
%O A208034 0,2
%A A208034 _Paul D. Hanna_, Feb 22 2012