A054329 One quarter of fourth unsigned column of Lanczos' triangle A053125.

1, 20, 224, 1920, 14080, 93184, 573440, 3342336, 18677760, 100925440, 530579456, 2726297600, 13740539904, 68115496960, 332859965440, 1606317768704, 7666516623360, 36232344109056, 169737107537920, 788899592929280

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. A054322, A053125.

Programs

GAP

List([0..30], n-> 4^(n-1)*Binomial(2*n+4,3)); # G. C. Greubel, Jul 22 2019

Magma

[4^(n-1)*Binomial(2*n+4,3): n in [0..30]]; // G. C. Greubel, Jul 22 2019

Mathematica

Table[4^(n-1)*Binomial[2*n+4, 3], {n,0,30}] (* G. C. Greubel, Jul 22 2019 *)

PARI

vector(30, n, n--; 4^(n-1)*binomial(2*n+4,3)) \\ G. C. Greubel, Jul 22 2019

Sage

[4^(n-1)*binomial(2*n+4,3) for n in (0..30)] # G. C. Greubel, Jul 22 2019

Formula

a(n)= 4^(n-1)*binomial(2*n+4, 3)= -A053125(n+3, 3)/4 = A054322(n)/4.

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

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