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A123027 Triangle of coefficients of (1 - x)^n*U(n,-(3*x - 2)/(2*x - 2)), where U(n,x) is the n-th Chebyshev polynomial of the second kind.

%I A123027 #23 Feb 16 2025 08:33:02
%S A123027 1,-2,3,3,-10,8,-4,22,-38,21,5,-40,111,-130,55,-6,65,-256,474,-420,
%T A123027 144,7,-98,511,-1324,1836,-1308,377,-8,140,-924,3130,-6020,6666,-3970,
%U A123027 987,9,-192,1554,-6588,16435,-25088,23109,-11822,2584,-10,255,-2472,12720,-39430,77645,-98160,77378,-34690,6765
%N A123027 Triangle of coefficients of (1 - x)^n*U(n,-(3*x - 2)/(2*x - 2)), where U(n,x) is the n-th Chebyshev polynomial of the second kind.
%C A123027 The n-th row consists of the coefficients in the expansion of Sum_{j=0..n} A053122(n,j)*x^j*(1 - x)^(n - j).
%H A123027 G. C. Greubel, <a href="/A123027/b123027.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A123027 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html">Chebyshev Polynomial of the Second Kind</a>
%H A123027 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev_polynomials">Chebyshev polynomials</a>
%F A123027 From _Franck Maminirina Ramaharo_, Oct 10 2018: (Start)
%F A123027 Row n = coefficients in the expansion of (1/sqrt((5*x - 4)*x))*(((3*x - 2 + sqrt((5*x - 4)*x))/2)^(n + 1) - ((3*x - 2 - sqrt((5*x - 4)*x))/2)^(n + 1)).
%F A123027 G.f.: 1/(1 + (2 - 3*x)*t + (1 - x)^2*t^2).
%F A123027 E.g.f.: exp(t*(3*x - 2)/2)*(sqrt((5*x - 4)*x)*cosh(t*sqrt((5*x - 4)*x)/2) + (3*x - 2)*sinh(t*sqrt((5*x - 4)*x)/2))/sqrt((5*x - 4)*x).
%F A123027 T(n,1) = (-1)^(n+1)*A006503(n).
%F A123027 T(n,n) = A001906(n+1). (End)
%e A123027 Triangle begins:
%e A123027      1;
%e A123027     -2,    3;
%e A123027      3,  -10,    8;
%e A123027     -4,   22,   38,    21;
%e A123027      5,  -40,  111,  -130,    55;
%e A123027     -6,   65, -256,   474,  -420,    144;
%e A123027      7,  -98,  511, -1324,  1836,  -1308,   377;
%e A123027     -8,  140, -924,  3130, -6020,   6666, -3970,    987;
%e A123027      9, -192, 1554, -6588, 16435, -25088, 23109, -11822, 2584;
%e A123027      ... reformatted and extended. _Franck Maminirina Ramaharo_, Oct 10 2018
%t A123027 b0 = Table[CoefficientList[ChebyshevU[n, x/2 -1], x], {n, 0, 10}];
%t A123027 Table[CoefficientList[Sum[b0[[m+1]][[n+1]]*x^n*(1-x)^(m-n), {n,0,m}], x], {m,0,10}]//Flatten
%t A123027 (* Alternative Adamson Matrix method *)
%t A123027 t[n_, m_] = If[n==m, 2, If[n==m-1 || n==m+1, 1, 0]];
%t A123027 M[d_] := Table[t[n, m], {n, d}, {m, d}];
%t A123027 a = Join[{{1}}, Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], {d, 1, 10}]];
%t A123027 Table[CoefficientList[Sum[a[[m+1]][[n+1]]*x^n*(1-x)^(m-n), {n, 0, m}], x], {m, 0, 10}]//Flatten
%o A123027 (Maxima)
%o A123027 A053122(n, k) := if n < k then 0 else ((-1)^(n - k))*binomial(n + k + 1,  2*k + 1)$
%o A123027 P(x, n) := expand(sum(A053122(n, j)*x^j*(1 - x)^(n - j), j, 0, n))$
%o A123027 T(n, k) := ratcoef(P(x, n), x, k)$
%o A123027 tabl(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, hipow(P(x, n), x)))$ /* _Franck Maminirina Ramaharo_, Oct 10 2018 */
%o A123027 (Sage)
%o A123027 def A053122(n, k): return 0 if (n<k) else (-1)^(n-k)*binomial(n+k+1, 2*k+1)
%o A123027 def p(n,x): return sum( A053122(n, j)*x^j*(1-x)^(n-j) for j in (0..n) )
%o A123027 def T(n): return ( p(n,x) ).full_simplify().coefficients(sparse=False)
%o A123027 flatten([T(n) for n in (0..12)]) # _G. C. Greubel_, Jul 15 2021
%Y A123027 Cf. A008310, A049310, A053122, A111006.
%Y A123027 Cf. A122753, A123018, A123019, A123021, A123199, A123202, A123217, A123221.
%K A123027 sign,tabl
%O A123027 0,2
%A A123027 _Roger L. Bagula_ and _Gary W. Adamson_, Sep 24 2006
%E A123027 Edited, new name, and offset corrected by _Franck Maminirina Ramaharo_, Oct 10 2018