A317451 a(n) = (n*A003500(n) - A231896(n))/2.

0, 2, 16, 92, 464, 2182, 9824, 42936, 183648, 772746, 3209968, 13196564, 53791408, 217700110, 875718080, 3504277360, 13959102912, 55383875346, 218965651152, 862983998924, 3391602170512, 13295446717334, 51999641009696, 202948920530728, 790569797639456, 3074179492922778

Offset: 0

Author

Rigoberto Florez, Jul 28 2018

Keywords

• nonn
• easy

Comments

Derivative of Chebyshev second kind polynomials evaluated at 2.

References

• 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.

Links

• Colin Barker, Table of n, a(n) for n = 0..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).

Crossrefs

Cf. A003500, A231896, A133156 (Chebyshev polynomials of the second kind).

Programs

Mathematica

CoefficientList[ Series[2 x/(x^2 - 4x + 1)^2, {x, 0, 25}], x] (* Robert G. Wilson v, Aug 07 2018 *)

PARI

a(n) = subst(deriv(polchebyshev(n, 2)), x, 2); \\ Michel Marcus, Jul 29 2018.

PARI

concat(0, Vec(2*x / (1 - 4*x + x^2)^2 + O(x^40))) \\ Colin Barker, Aug 06 2018

Formula

From Colin Barker, Aug 06 2018: (Start)

G.f.: 2*x / (1 - 4*x + x^2)^2.

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.

a(n) = 8*a(n-1) - 18*a(n-2) + 8*a(n-3) - a(n-4) for n>3.

(End)