A317405 a(n) = n * A001353(n).
1, 8, 45, 224, 1045, 4680, 20377, 86912, 364905, 1513160, 6211909, 25290720, 102251773, 410963336, 1643288625, 6541692416, 25939798993, 102503274120, 403800061789, 1586318259680, 6216231359205, 24304019419592, 94826736906697, 369285078314880, 1435615286196025
Offset: 1
Links
- Colin Barker, Table of n, a(n) for n = 1..1000
- Rigoberto Flórez, Robinson Higuita, and Alexander Ramírez, The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials, arXiv:1808.01264 [math.NT], 2018.
- Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Star of David and other patterns in the Hosoya-like polynomials triangles, Journal of Integer Sequences, Vol. 21 (2018), Article 18.4.6.
- R. Flórez, N. McAnally, and A. Mukherjees, Identities for the generalized Fibonacci polynomial, Integers, 18B (2018), Paper No. A2.
- R. Flórez, R. Higuita and A. Mukherjees, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.
- Eric Weisstein's World of Mathematics, Chebyshev Polynomial of the First Kind
- Index entries for linear recurrences with constant coefficients, signature (8,-18,8,-1).
Programs
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Mathematica
Table[ D[ ChebyshevT[n, x], x] /. x -> 2, {n, 25}] CoefficientList[Series[-x(x^2 - 1)/(x^2 - 4x + 1)^2, {x, 0, 24}], x] (* Robert G. Wilson v, Aug 07 2018 *) -
PARI
Vec(x*(1 - x)*(1 + x) / (1 - 4*x + x^2)^2 + O(x^40)) \\ Colin Barker, Jul 28 2018
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PARI
a(n) = subst(deriv(polchebyshev(n)), x, 2); \\ Michel Marcus, Jul 29 2018
Formula
From Colin Barker, Jul 28 2018: (Start)
G.f.: x*(1 - x)*(1 + x) / (1 - 4*x + x^2)^2.
a(n) = (((-(2-sqrt(3))^n + (2+sqrt(3))^n)*n)) / (2*sqrt(3)).
a(n) = 8*a(n-1) - 18*a(n-2) + 8*a(n-3) - a(n-4) for n>4.
(End)
Comments