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%I #53 Sep 08 2022 08:46:01
%S 1,15,128,816,4320,20064,84480,329472,1208064,4209920,14057472,
%T 45260800,141213696,428654592,1270087680,3683254272,10478223360,
%U 29297934336,80648077312,218864025600,586290298880,1551944908800,4063273943040
%N Negated coefficients of Chebyshev T polynomials: a(n) = -A053120(n+14, n), n >= 0.
%C 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.
%H G. C. Greubel, <a href="/A209404/b209404.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (16,-112,448,-1120,1792,-1792,1024,-256).
%F 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]
%F G.f.: (1-x)/(1-2*x)^8. - _Colin Barker_, May 31 2013
%F 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
%p seq(2^(n-1)*(n+14)*binomial(n+6,6)/7, n=0..30); # _G. C. Greubel_, Oct 18 2019
%t 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 *)
%o (Magma) Rn:=6; [2^(n-1)/(Rn+1)*Binomial(n+Rn,Rn)*(n+(Rn+1)*2) : n in [0..22]];
%o (PARI) for(n=0,30, print1(2^(n-1)*binomial(n+6,6)*(n+14)/7, ", ")) \\ _G. C. Greubel_, Jan 03 2018
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 23); Coefficients(R!( (1-x)/(1-2*x)^8 )); // _Marius A. Burtea_, Oct 17 2019
%o (GAP) List([0..30], n-> 2^(n-1)*(n+14)*Binomial(n+6,6)/7); # _G. C. Greubel_, Oct 18 2019
%Y Cf. A001792, A001793, A001794, A006974, A006975, A006976, A053120.
%K nonn,easy
%O 0,2
%A _Brad Clardy_, Mar 08 2012
%E Name made more specific by _Wolfdieter Lang_, Nov 25 2019
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