A209404 Negated coefficients of Chebyshev T polynomials: a(n) = -A053120(n+14, n), n >= 0.

1, 15, 128, 816, 4320, 20064, 84480, 329472, 1208064, 4209920, 14057472, 45260800, 141213696, 428654592, 1270087680, 3683254272, 10478223360, 29297934336, 80648077312, 218864025600, 586290298880, 1551944908800, 4063273943040

Offset: 0

Author

Brad Clardy, Mar 08 2012

Keywords

• nonn
• easy

Comments

The MAGMA program provided calculates the coefficients of order one Chebyshev polynomials, for any arbitrary level. For example, setting Rn to 0 produces A001792, 1 produces A001793, 2 produces A001794, 3 produces A006974, 4 produces A006975, and 5 produces A006976. This sequence is produced with an Rn of 6.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (16,-112,448,-1120,1792,-1792,1024,-256).

Crossrefs

Cf. A001792, A001793, A001794, A006974, A006975, A006976, A053120.

Programs

GAP

List([0..30], n-> 2^(n-1)*(n+14)*Binomial(n+6,6)/7); # G. C. Greubel, Oct 18 2019

Magma

Rn:=6; [2^(n-1)/(Rn+1)*Binomial(n+Rn,Rn)*(n+(Rn+1)*2) : n in [0..22]];

Magma

R:=PowerSeriesRing(Integers(), 23); Coefficients(R!( (1-x)/(1-2*x)^8 )); // Marius A. Burtea, Oct 17 2019

Maple

seq(2^(n-1)*(n+14)*binomial(n+6,6)/7, n=0..30); # G. C. Greubel, Oct 18 2019

Mathematica

CoefficientList[Series[(1-x)/(1-2*x)^8, {x,0,30}], x] (* or *) Table[2^(n-1)*Binomial[n+6,6]*(n+14)/7, {n,0,30}] (* G. C. Greubel, Jan 03 2018 *)

PARI

for(n=0,30, print1(2^(n-1)*binomial(n+6,6)*(n+14)/7, ", ")) \\ G. C. Greubel, Jan 03 2018

Formula

a(n) = 2^(n-1)*binomial(n+6, 6)*(n+14)/7 = -A053120(n+14, n), n >= 0. [See a comment in A053120 on subdiagonal sequences. - Wolfdieter Lang, Jan 03 2020]

G.f.: (1-x)/(1-2*x)^8. - Colin Barker, May 31 2013

E.g.f.: (1/315)*exp(2*x)*(315 + 4095*x + 11340*x^2 + 11550*x^3 + 5250*x^4 + 1134*x^5 + 112*x^6 + 4*x^7). - Stefano Spezia, Oct 17 2019

Extensions

Name made more specific by Wolfdieter Lang, Nov 25 2019