A002700 Coefficients of Chebyshev polynomials: n*(2*n+1) * 4^(n-1).

3, 40, 336, 2304, 14080, 79872, 430080, 2228224, 11206656, 55050240, 265289728, 1258291200, 5888802816, 27246198784, 124822487040, 566935683072, 2555505541120, 11441792876544, 50921132261376, 225399883694080, 992858999881728, 4354066045992960

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• Cornelius Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 518.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Cornelius Lanczos, Applied Analysis. (Annotated scans of selected pages)
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
• Index entries for linear recurrences with constant coefficients, signature (12,-48,64).
• Index entries for sequences related to Chebyshev polynomials.

Crossrefs

Cf. A002699.

Programs

GAP

List([1..30], n-> 4^(n-1)*n*(2*n+1)); # G. C. Greubel, Jul 23 2019

Magma

[4^(n-1)*n*(2*n+1): n in [1..30]]; // G. C. Greubel, Jul 23 2019

Maple

A002700:=-(3+4*z)/(4*z-1)**3; # Simon Plouffe in his 1992 dissertation.

Mathematica

Table[n*(2*n+1)*2^(2*n-2),{n,1,30}] (* Vaclav Kotesovec, Jun 03 2014 *)
LinearRecurrence[{12,-48,64},{3,40,336},30] (* Harvey P. Dale, May 17 2018 *)

PARI

Vec(-x*(4*x+3)/(4*x-1)^3 + O(x^30)) \\ Colin Barker, Jun 15 2015

Sage

[4^(n-1)*n*(2*n+1) for n in (1..30)] # G. C. Greubel, Jul 23 2019

Formula

a(n) = 12*a(n-1) - 48*a(n-2) + 64*a(n-3). - Colin Barker, Jun 15 2015

a(n) = 1/2*Sum_{k = 0..2*n} k^2*binomial(2*n,k). Cf. A002699. - Peter Bala, Apr 09 2017

From Amiram Eldar, Feb 17 2023: (Start)

Sum_{n>=1} 1/a(n) = 8 + 8*log(2) - 12*log(3).

Sum_{n>=1} (-1)^(n+1)/a(n) = 16*arctan(1/2) + 4*log(5/4) - 8. (End)