A054322 Fourth unsigned column of Lanczos triangle A053125 (decreasing powers).

4, 80, 896, 7680, 56320, 372736, 2293760, 13369344, 74711040, 403701760, 2122317824, 10905190400, 54962159616, 272461987840, 1331439861760, 6425271074816, 30666066493440, 144929376436224, 678948430151680

Offset: 0

Author

Wolfdieter Lang

Keywords

• easy
• nonn

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 (16,-96,256,-256).

Crossrefs

Cf. A053125, A053123, A002700, A054329.

Programs

GAP

List([0..20], n-> 4^n*Binomial(2*n+4, 3)); # G. C. Greubel, Jul 22 2019

Magma

[4^n*Binomial(2*n+4, 3): n in [0..20]]; // G. C. Greubel, Jul 22 2019

Mathematica

Table[4^n*Binomial[2*n+4, 3], {n,0,20}] (* G. C. Greubel, Jul 22 2019 *)
LinearRecurrence[{16,-96,256,-256},{4,80,896,7680},20] (* Harvey P. Dale, Mar 27 2023 *)

PARI

vector(20, n, n--; 4^n*binomial(2*n+4, 3)) \\ G. C. Greubel, Jul 22 2019

Sage

[4^n*binomial(2*n+4, 3) for n in (0..20)] # G. C. Greubel, Jul 22 2019

Formula

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

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

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