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

1,2

Comments

Derivative of Chebyshev polynomials of the first kind evaluated at x=2.

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).

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)

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 *)

Prog

(PARI) Vec(x*(1 - x)*(1 + x) / (1 - 4*x + x^2)^2 + O(x^40)) \\ Colin Barker, Jul 28 2018
(PARI) a(n) = subst(deriv(polchebyshev(n)), x, 2); \\ Michel Marcus, Jul 29 2018

Crossrefs

Cf. A001353, A028297 (Chebyshev polynomials of the first kind).

Sequence in context: A016208 A216540 A026852 * A110609 A201190 A297089

Adjacent sequences: A317402 A317403 A317404 * A317406 A317407 A317408

Keyword

nonn,easy

Author

Rigoberto Florez, Jul 27 2018

Status

approved