|
%I M3147 N1275 #50 Feb 17 2023 08:15:47
%S 3,40,336,2304,14080,79872,430080,2228224,11206656,55050240,265289728,
%T 1258291200,5888802816,27246198784,124822487040,566935683072,
%U 2555505541120,11441792876544,50921132261376,225399883694080,992858999881728,4354066045992960
%N Coefficients of Chebyshev polynomials: n*(2*n+1) * 4^(n-1).
%D Cornelius Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 518.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Colin Barker, <a href="/A002700/b002700.txt">Table of n, a(n) for n = 1..1000</a>
%H Cornelius Lanczos, <a href="/A002457/a002457.pdf">Applied Analysis</a>. (Annotated scans of selected pages)
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (12,-48,64).
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%F a(n) = 12*a(n-1) - 48*a(n-2) + 64*a(n-3). - _Colin Barker_, Jun 15 2015
%F a(n) = 1/2*Sum_{k = 0..2*n} k^2*binomial(2*n,k). Cf. A002699. - _Peter Bala_, Apr 09 2017
%F From _Amiram Eldar_, Feb 17 2023: (Start)
%F Sum_{n>=1} 1/a(n) = 8 + 8*log(2) - 12*log(3).
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 16*arctan(1/2) + 4*log(5/4) - 8. (End)
%p A002700:=-(3+4*z)/(4*z-1)**3; # _Simon Plouffe_ in his 1992 dissertation.
%t Table[n*(2*n+1)*2^(2*n-2),{n,1,30}] (* _Vaclav Kotesovec_, Jun 03 2014 *)
%t LinearRecurrence[{12,-48,64},{3,40,336},30] (* _Harvey P. Dale_, May 17 2018 *)
%o (PARI) Vec(-x*(4*x+3)/(4*x-1)^3 + O(x^30)) \\ _Colin Barker_, Jun 15 2015
%o (Magma) [4^(n-1)*n*(2*n+1): n in [1..30]]; // _G. C. Greubel_, Jul 23 2019
%o (Sage) [4^(n-1)*n*(2*n+1) for n in (1..30)] # _G. C. Greubel_, Jul 23 2019
%o (GAP) List([1..30], n-> 4^(n-1)*n*(2*n+1)); # _G. C. Greubel_, Jul 23 2019
%Y Cf. A002699.
%K nonn,easy
%O 1,1
%A _N. J. A. Sloane_
|
