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A100551 Coefficient list of ChebyshevU(n, 1-x).

%I A100551 #14 Mar 27 2023 05:16:33
%S A100551 1,2,-2,3,-8,4,4,-20,24,-8,5,-40,84,-64,16,6,-70,224,-288,160,-32,7,
%T A100551 -112,504,-960,880,-384,64,8,-168,1008,-2640,3520,-2496,896,-128,9,
%U A100551 -240,1848,-6336,11440,-11648,6720,-2048,256,10,-330,3168,-13728,32032,-43680,35840,-17408,4608,-512
%N A100551 Coefficient list of ChebyshevU(n, 1-x).
%H A100551 G. C. Greubel, <a href="/A100551/b100551.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A100551 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A100551 G.f.: ChebyshevU(n, 1-x).
%F A100551 From _G. C. Greubel_, Mar 27 2023: (Start)
%F A100551 T(n, k) = binomial(n+k+1, n-k)*(-2)^k.
%F A100551 T(n, n) = A122803(n).
%F A100551 T(n, n-1) = 2*(-1)^(n-1)*A001787(n), n >= 1.
%F A100551 Sum_{k=0..n} T(n, k) = A056594(n).
%F A100551 Sum_{k=0..n} (-1)^k*T(n, k) = A001353(n+1). (End)
%e A100551 Triangle begins as:
%e A100551   1;
%e A100551   2,   -2;
%e A100551   3,   -8,    4;
%e A100551   4,  -20,   24,    -8;
%e A100551   5,  -40,   84,   -64,    16;
%e A100551   6,  -70,  224,  -288,   160,    -32;
%e A100551   7, -112,  504,  -960,   880,   -384,   64;
%e A100551   8, -168, 1008, -2640,  3520,  -2496,  896,  -128;
%e A100551   9, -240, 1848, -6336, 11440, -11648, 6720, -2048, 256;
%t A100551 Table[CoefficientList[ChebyshevU[n, 1-x], x], {n, 0, 12}]
%o A100551 (PARI) row(n) = Vecrev(polchebyshev(n, 2, 1-x)); \\ _Michel Marcus_, Apr 27 2020
%o A100551 (Magma) [Binomial(n+k+1, n-k)*(-2)^k: k in [0..n], n in [0..12]]; // _G. C. Greubel_, Mar 27 2023
%o A100551 (SageMath)
%o A100551 def A100551(n,k): return binomial(n+k+1, n-k)*(-2)^k
%o A100551 flatten([[A100551(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Mar 27 2023
%Y A100551 Cf. A001353, A001787, A053117, A056594, A122803.
%K A100551 easy,sign,tabl
%O A100551 0,2
%A A100551 _Wouter Meeussen_, Nov 27 2004
%E A100551 Keyword tabl from _Michel Marcus_, Apr 27 2020