A054331 One eighth of eighth unsigned column of Lanczos triangle A053125.

1, 60, 1584, 27456, 366080, 4073472, 39690240, 349274112, 2835283968, 21554790400, 155194490880, 1067345510400, 7058711642112, 45127489814528, 280101660917760, 1693862087098368, 10009185060126720, 57935518230380544, 329171172648288256, 1839094001030922240, 10119502600794537984

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 (32,-448,3584,-17920,57344,-114688,131072,-65536).

Index entries for sequences related to Chebyshev polynomials.

Formula

a(n) = 2^(2*n-3)*binomial(2*n+8, 7) = -A053125(n+7, 7)/8 = A054326(n)/8.

G.f.: (1+4*x)*(1+24*x+16*x^2)/(1-4*x)^8.

From Amiram Eldar, Oct 30 2025: (Start)

Sum_{n>=0} 1/a(n) = 165144/5 + 81760*log(2) - 81648*log(3).

Sum_{n>=0} (-1)^n/a(n) = -37464/5 + 9856*arctan(1/2) + 13104*log(5/4). (End)

Mathematica

Table[4^n Binomial[2n+8, 7]/8, {n, 0, 20}] (* Harvey P. Dale, Nov 03 2011 *)
LinearRecurrence[{32, -448, 3584, -17920, 57344, -114688, 131072, -65536}, {1, 60, 1584, 27456, 366080, 4073472, 39690240, 349274112}, 20] (* Harvey P. Dale, Feb 25 2022 *)

Prog

(PARI) vector(20, n, n--; 2^(2*n-3)*binomial(2*n+8, 7)) \\ G. C. Greubel, Jul 22 2019
(Magma) [2^(2*n-3)*Binomial(2*n+8, 7): n in [0..20]]; // G. C. Greubel, Jul 22 2019
(SageMath) [2^(2*n-3)*binomial(2*n+8, 7) for n in (0..20)] # G. C. Greubel, Jul 22 2019
(GAP) List([0..20], n-> 2^(2*n-3)*Binomial(2*n+8, 7)); # G. C. Greubel, Jul 22 2019

Crossrefs

Cf. A054326, A053125.

Sequence in context: A269196 A379607 A393526 * A160349 A281373 A053528

Adjacent sequences: A054328 A054329 A054330 * A054332 A054333 A054334

Keyword

easy,nice,nonn

Author

Wolfdieter Lang

Status

approved