A054323 - OEIS

%I #19 Sep 08 2022 08:45:00

%S 5,140,2016,21120,183040,1397760,9748480,63504384,392232960,

%T 2321285120,13264486400,73610035200,398475657216,2111580405760,

%U 10984378859520,56221121904640,283661115064320,1413061420253184,6959221409054720

%N Fifth column of Lanczos triangle A053125 (decreasing powers).

%D C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 518.

%D Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

%H G. C. Greubel, <a href="/A054323/b054323.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (20, -160, 640, -1280, 1024).

%F a(n) = 4^n*binomial(2*n+5, 4) = 4^n*A053126(n+4) = A053125(n+4, 4).

%F G.f.: (5 +40*x +16*x^2)/(1-4*x)^5.

%F E.g.f.: (15 +360*x +1464*x^2 +1664*x^3 +512*x^4)*exp(4*x)/3. - _G. C. Greubel_, Jul 22 2019

%F a(n) = 20*a(n-1)-160*a(n-2)+640*a(n-3)-1280*a(n-4)+1024*a(n-5). - _Wesley Ivan Hurt_, May 02 2021

%t Table[4^n Binomial[2n+5,4],{n,0,20}] (* or *) LinearRecurrence[{20,-160, 640,-1280,1024},{5,140,2016,21120,183040},20] (* _Harvey P. Dale_, Mar 03 2018 *)

%o (PARI) vector(20, n, n--; 4^n*binomial(2*n+5, 4)) \\ _G. C. Greubel_, Jul 22 2019

%o (Magma) [4^n*Binomial(2*n+5, 4): n in [0..20]]; // _G. C. Greubel_, Jul 22 2019

%o (Sage) [4^n*binomial(2*n+5, 4) for n in (0..20)] # _G. C. Greubel_, Jul 22 2019

%o (GAP) List([0..20], n-> 4^n*Binomial(2*n+5, 4)); # _G. C. Greubel_, Jul 22 2019

%Y Cf. A053125, A054322.

%Y Bisection of A080951.

%K nonn,easy

%O 0,1

%A _Wolfdieter Lang_