A054325 Seventh column of Lanczos triangle A053125 (decreasing powers).

7, 336, 7392, 109824, 1281280, 12673024, 111132672, 889061376, 6615662592, 46425702400, 310388981760, 1992378286080, 12352745373696, 74327630282752, 435713694760960, 2496217812566016, 14012859084177408, 77247357640507392, 418945128825094144, 2238897044733296640

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 (28,-336,2240,-8960,21504,-28672,16384).

Index entries for sequences related to Chebyshev polynomials.

Formula

a(n) = 4^n*binomial(2*n+7, 6) = A053125(n+6, 6).

G.f.: (7 +140*x +336*x^2 +64*x^3)/(1-4*x)^7.

From Amiram Eldar, Oct 30 2025: (Start)

Sum_{n>=0} 1/a(n) = 5954/5 + 2904*log(2) - 2916*log(3).

Sum_{n>=0} (-1)^n/a(n) = 1566/5 - 912*arctan(1/2) + 492*log(5/4). (End)

Mathematica

Table[4^n*Binomial[2*n+7, 6], {n, 0, 20}] (* G. C. Greubel, Jul 22 2019 *)

Prog

(PARI) vector(20, n, n--; 4^n*binomial(2*n+7, 6)) \\ G. C. Greubel, Jul 22 2019
(Magma) [4^n*Binomial(2*n+7, 6): n in [0..20]]; // G. C. Greubel, Jul 22 2019
(SageMath) [4^n*binomial(2*n+7, 6) for n in (0..20)] # G. C. Greubel, Jul 22 2019
(GAP) List([0..20], n-> 4^n*Binomial(2*n+7, 6)); # G. C. Greubel, Jul 22 2019

Crossrefs

Cf. A053125, A054324.

Sequence in context: A324233 A009587 A281619 * A161582 A200967 A068150

Adjacent sequences: A054322 A054323 A054324 * A054326 A054327 A054328

Keyword

nonn,easy

Author

Wolfdieter Lang

Status

approved