A054323 Fifth column of Lanczos triangle A053125 (decreasing powers).

5, 140, 2016, 21120, 183040, 1397760, 9748480, 63504384, 392232960, 2321285120, 13264486400, 73610035200, 398475657216, 2111580405760, 10984378859520, 56221121904640, 283661115064320, 1413061420253184, 6959221409054720

Offset: 0

Author

Wolfdieter Lang

Keywords

• nonn
• easy

References

• C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 518.
• Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (20, -160, 640, -1280, 1024).

Crossrefs

Cf. A053125, A054322.

Bisection of A080951.

Programs

GAP

List([0..20], n-> 4^n*Binomial(2*n+5, 4)); # G. C. Greubel, Jul 22 2019

Magma

[4^n*Binomial(2*n+5, 4): n in [0..20]]; // G. C. Greubel, Jul 22 2019

Mathematica

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 *)

PARI

vector(20, n, n--; 4^n*binomial(2*n+5, 4)) \\ G. C. Greubel, Jul 22 2019

Sage

[4^n*binomial(2*n+5, 4) for n in (0..20)] # G. C. Greubel, Jul 22 2019

Formula

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

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

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

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