A054324 Sixth unsigned column of Lanczos triangle A053125 (decreasing powers).

6, 224, 4032, 50688, 512512, 4472832, 35094528, 254017536, 1725825024, 11142168576, 68975329280, 412216197120, 2390853943296, 13514114596864, 74693776244736, 404792077713408, 2155824474488832, 11304491362025472, 58457459836059648, 298519605964439552, 1507159961820463104

Offset

0,1

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 (24,-240,1280,-3840,6144,-4096).

Index entries for sequences related to Chebyshev polynomials.

Formula

a(n) = 4^n*binomial(2*n+6, 5) = -A053125(n+5, 5)= 2*A054330(n).

G.f.: 2*(3+4*x)*(1+12*x)/(1-4*x)^6.

E.g.f.: (2/15)*(45 +1500*x +8760*x^2 +15840*x^3 +10240*x^4 +2048*x^5) * exp(4*x). - G. C. Greubel, Jul 22 2019

From Amiram Eldar, Oct 30 2025: (Start)

Sum_{n>=0} 1/a(n) = 965/3 + 820*log(2) - 810*log(3).

Sum_{n>=0} (-1)^n/a(n) = 715/3 - 480*arctan(1/2) - 70*log(5/4). (End)

Mathematica

Table[4^n Binomial[2n+6, 5], {n, 0, 20}] (* or *) LinearRecurrence[{24, -240, 1280, -3840, 6144, -4096}, {6, 224, 4032, 50688, 512512, 4472832}, 20] (* Harvey P. Dale, Jul 02 2017 *)

Prog

(PARI) vector(20, n, n--; 4^n*binomial(2*n+6, 5)) \\ G. C. Greubel, Jul 22 2019
(Magma) [4^n*Binomial(2*n+6, 5): n in [0..20]]; // G. C. Greubel, Jul 22 2019
(SageMath) [4^n*binomial(2*n+6, 5) for n in (0..20)] # G. C. Greubel, Jul 22 2019
(GAP) List([0..20], n-> 4^n*Binomial(2*n+6, 5)); # G. C. Greubel, Jul 22 2019

Crossrefs

Cf. A053125, A054323, A054330.

Sequence in context: A144045 A355669 A061610 * A117255 A130644 A223100

Adjacent sequences: A054321 A054322 A054323 * A054325 A054326 A054327

Keyword

nonn,easy

Author

Wolfdieter Lang

Status

approved