%I #13 Mar 27 2023 10:19:09

%S 4,80,896,7680,56320,372736,2293760,13369344,74711040,403701760,

%T 2122317824,10905190400,54962159616,272461987840,1331439861760,

%U 6425271074816,30666066493440,144929376436224,678948430151680

%N Fourth unsigned 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="/A054322/b054322.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_04">Index entries for linear recurrences with constant coefficients</a>, signature (16,-96,256,-256).

%F a(n) = 4^n*binomial(2*n+4, 3) = -A053125(n+3, 3) = 4*A054329(n).

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

%F E.g.f.: (4/3)*(3 +48*x +120*x^2 +64*x^3)*exp(4*x). - _G. C. Greubel_, Jul 22 2019

%t Table[4^n*Binomial[2*n+4, 3], {n,0,20}] (* _G. C. Greubel_, Jul 22 2019 *)

%t LinearRecurrence[{16,-96,256,-256},{4,80,896,7680},20] (* _Harvey P. Dale_, Mar 27 2023 *)

%o (PARI) vector(20, n, n--; 4^n*binomial(2*n+4, 3)) \\ _G. C. Greubel_, Jul 22 2019

%o (Magma) [4^n*Binomial(2*n+4, 3): n in [0..20]]; // _G. C. Greubel_, Jul 22 2019

%o (Sage) [4^n*binomial(2*n+4, 3) for n in (0..20)] # _G. C. Greubel_, Jul 22 2019

%o (GAP) List([0..20], n-> 4^n*Binomial(2*n+4, 3)); # _G. C. Greubel_, Jul 22 2019

%Y Cf. A053125, A053123, A002700, A054329.

%K easy,nonn

%O 0,1

%A _Wolfdieter Lang_