A054327 Ninth column of Lanczos triangle A053125 (decreasing powers).

9, 660, 20592, 411840, 6223360, 77395968, 833495040, 8033304576, 70882099200, 581979340800, 4500640235520, 33087710822400, 232937484189696, 1579462143508480, 10363761453957120, 66060621396836352, 410376587465195520, 2491227283906363392, 14812702769172971520, 86437418048453345280

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 (36,-576,5376,-32256,129024,-344064,589824,-589824,262144).

Index entries for sequences related to Chebyshev polynomials.

Formula

a(n) = 4^n*binomial(2*n+9, 8) = A053125(n+8, 8).

G.f.: (4*x+3)*(64*x^3+528*x^2+108*x+3)/(1-4*x)^9.

From Amiram Eldar, Oct 30 2025: (Start)

Sum_{n>=0} 1/a(n) = 298184/21 + 34976*log(2) - 34992*log(3).

Sum_{n>=0} (-1)^n/a(n) = -443944/105 + 8896*arctan(1/2) + 464*log(5/4). (End)

Mathematica

Table[4^n*Binomial[2*n+9, 8], {n, 0, 20}] (* G. C. Greubel, Jul 22 2019 *)

Prog

(PARI) vector(20, n, n--; 4^n*binomial(2*n+9, 8)) \\ G. C. Greubel, Jul 22 2019
(Magma) [4^n*Binomial(2*n+9, 8): n in [0..20]]; // G. C. Greubel, Jul 22 2019
(SageMath) [4^n*binomial(2*n+9, 8) for n in (0..20)] # G. C. Greubel, Jul 22 2019
(GAP) List([0..20], n-> 4^n*Binomial(2*n+9, 8)); # G. C. Greubel, Jul 22 2019

Crossrefs

Cf. A053125, A054326.

Sequence in context: A091062 A221133 A015007 * A209964 A287937 A372184

Adjacent sequences: A054324 A054325 A054326 * A054328 A054329 A054330

Keyword

nonn,easy

Author

Wolfdieter Lang

Status

approved