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, 3640482999566336, 16690586509639680, 76068612456054784

Offset

0,2

References

Cornelius Lanczos, Applied Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 518.

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (16,-96,256,-256).

Index entries for sequences related to Chebyshev polynomials.

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

From Amiram Eldar, Oct 30 2025: (Start)

Sum_{n>=0} 1/a(n) = 72 - 216*log(3) + 240*log(2).

Sum_{n>=0} (-1)^n/a(n) = 192*arctan(1/2) - 72*(log(5/4)+1). (End)

Mathematica

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

Prog

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

Crossrefs

Cf. A054322, A053125.

Sequence in context: A000833 A302661 A178261 * A112503 A007160 A195265

Adjacent sequences: A054326 A054327 A054328 * A054330 A054331 A054332

Keyword

easy,nonn

Author

Wolfdieter Lang

Status

approved