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%I #34 Apr 25 2019 13:30:13
%S 0,2,16,92,464,2182,9824,42936,183648,772746,3209968,13196564,
%T 53791408,217700110,875718080,3504277360,13959102912,55383875346,
%U 218965651152,862983998924,3391602170512,13295446717334,51999641009696,202948920530728,790569797639456,3074179492922778
%N a(n) = (n*A003500(n) - A231896(n))/2.
%C Derivative of Chebyshev second kind polynomials evaluated at 2.
%D R. Flórez, N. McAnally, and A. Mukherjees, Identities for the generalized Fibonacci polynomial, Integers, 18B (2018), Paper No. A2.
%D R. Flórez, R. Higuita and A. Mukherjees, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.
%H Colin Barker, <a href="/A317451/b317451.txt">Table of n, a(n) for n = 0..1000</a>
%H Rigoberto Flórez, Robinson Higuita, and Alexander Ramírez, <a href="https://arxiv.org/abs/1808.01264"> The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials</a>, arXiv:1808.01264 [math.NT], 2018.
%H Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Florez2/florez8.html">Star of David and other patterns in the Hosoya-like polynomials triangles</a>, Journal of Integer Sequences, Vol. 21 (2018), Article 18.4.6.
%H R. Flórez, N. McAnally, and A. Mukherjees, <a href="http://math.colgate.edu/~integers/s18b2/s18b2.Abstract.html">Identities for the generalized Fibonacci polynomial</a>, Integers, 18B (2018), Paper No. A2.
%H R. Flórez, R. Higuita and A. Mukherjees, <a href="http://math.colgate.edu/~integers/s14/s14.Abstract.html">Characterization of the strong divisibility property for generalized Fibonacci polynomials</a>, Integers, 18 (2018), Paper No. A14.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html">Chebyshev Polynomial of the First Kind</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-18,8,-1).
%F From _Colin Barker_, Aug 06 2018: (Start)
%F G.f.: 2*x / (1 - 4*x + x^2)^2.
%F a(n) = (sqrt(3)*((2-sqrt(3))^n - (2+sqrt(3))^n) + 3*((2-sqrt(3))^(1+n) + (2+sqrt(3))^(1+n))*n) / 18.
%F a(n) = 8*a(n-1) - 18*a(n-2) + 8*a(n-3) - a(n-4) for n>3.
%F (End)
%t CoefficientList[ Series[2 x/(x^2 - 4x + 1)^2, {x, 0, 25}], x] (* _Robert G. Wilson v_, Aug 07 2018 *)
%o (PARI) a(n) = subst(deriv(polchebyshev(n, 2)), x, 2); \\ _Michel Marcus_, Jul 29 2018.
%o (PARI) concat(0, Vec(2*x / (1 - 4*x + x^2)^2 + O(x^40))) \\ _Colin Barker_, Aug 06 2018
%Y Cf. A003500, A231896, A133156 (Chebyshev polynomials of the second kind).
%K nonn,easy
%O 0,2
%A _Rigoberto Florez_, Jul 28 2018
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