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

Offset: 0

Author

Wolfdieter Lang

Keywords

• easy
• nice
• nonn

References

• C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 518.
• Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (32,-448,3584,-17920,57344,-114688,131072,-65536).

Crossrefs

Cf. A054326, A053125.

Programs

GAP

List([0..20], 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

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 *)

PARI

vector(20, n, n--; 2^(2*n-3)*binomial(2*n+8, 7)) \\ G. C. Greubel, Jul 22 2019

Sage

[2^(2*n-3)*binomial(2*n+8, 7) for n in (0..20)] # G. C. Greubel, Jul 22 2019

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.