This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A272199 #37 Mar 03 2024 10:14:43
%S A272199 1,2,-9,-44,29,630,883,-6424,-24327,34858,385967,318780,-4380011,
%T A272199 -12904162,31131819,230017744,55321841,-2879586990,-6478357913,
%U A272199 24477915044,133174482957,-51863929658,-1834996137757,-2995761189960,17863427410921,74671750291322
%N A272199 Expansion of 1/(1 - 2*x + 13*x^2).
%C A272199 a(n) gives the coefficient c(13^n) of (eta(z^6))^4, a modular cusp form of weight 2, when expanded in powers of q = exp(2*Pi*i*z), Im(z) > 0, assuming alpha-multiplicativity (not valid for p = 2 and 3) with alpha(x) = x (weight 2) and input c(13) = +2. Eta is the Dedekind function. See the Apostol reference, p. 138, eq. (54) for alpha-multiplicativity and p. 130, eq. (39) with k=2. See also A000727 where a(n)=c(13^n) = A000727((13^n-1)/6)=A000727(2*A091030(n)), n >= 0. For the proof that alpha-multiplicativity leads to the recurrence involving Chebyshev's S polynomials see a comment on A168175, and the Apostol reference, Exercise 6., p. 139.
%D A272199 Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second edition, Springer, 1990, pp. 130, 138 - 139.
%H A272199 Indranil Ghosh, <a href="/A272199/b272199.txt">Table of n, a(n) for n = 0..1790</a>
%H A272199 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-13).
%H A272199 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A272199 G.f.: 1/(1 - 2*x + 13*x^2).
%F A272199 a(n) = 2*a(n-1) - 13*a(n-2), a(-1) = 0, a(0) = 1.
%F A272199 a(n) = sqrt(13)^n * S(n, 2/sqrt(13)), n >= 0, with Chebyshev's S polynomials (A049310).
%F A272199 a(n) = (Ap^(n+1) - Am^(n+1))/(Ap - Am) with Ap:= 1 + 2*sqrt(3)*i and Am = 1 - 2*sqrt(3)*i, (Binet - de Moivre formula), where i is the imaginary unit.
%F A272199 E.g.f.: (sqrt(3)*sin(2*sqrt(3)*x) + 6*cos(2*sqrt(3)*x))*exp(x)/6. - _Ilya Gutkovskiy_, Apr 27 2016
%e A272199 a(2) = c(13^2) = A000727(2*A091030(2)) =
%e A272199 A000727(28) = -9.
%t A272199 CoefficientList[Series[1/(1 - 2 x + 13 x^2), {x, 0, 25}], x] (* _Michael De Vlieger_, Apr 27 2016 *)
%t A272199 LinearRecurrence[{2, -13}, {1, 2}, 30] (* _Vincenzo Librandi_, Nov 25 2016 *)
%o A272199 (PARI) Vec(1/(1-2*x+13*x^2) + O(x^99)) \\ _Altug Alkan_, Apr 28 2016
%o A272199 (Magma) I:=[1,2]; [n le 2 select I[n] else 2*Self(n-1)-13*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Nov 25 2016
%Y A272199 Cf. A023000, A049310, A168175.
%K A272199 sign,easy
%O A272199 0,2
%A A272199 _Wolfdieter Lang_, Apr 27 2016