This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A053123 #45 Apr 19 2025 06:55:30
%S A053123 1,1,-2,1,-4,3,1,-6,10,-4,1,-8,21,-20,5,1,-10,36,-56,35,-6,1,-12,55,
%T A053123 -120,126,-56,7,1,-14,78,-220,330,-252,84,-8,1,-16,105,-364,715,-792,
%U A053123 462,-120,9,1,-18,136,-560,1365,-2002,1716,-792,165,-10,1,-20,171,-816,2380,-4368,5005,-3432,1287,-220,11,1
%N A053123 Triangle of coefficients of shifted Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in decreasing order).
%C A053123 T(n,m) = A053122(n,n-m).
%C A053123 G.f. for row polynomials and row sums same as in A053122.
%C A053123 Unsigned column sequences are A000012, A005843, A014105, A002492 for m=0..3, resp. and A053126-A053131 for m=4..9.
%C A053123 This is also the coefficient triangle for Chebyshev's U(2*n+1,x) polynomials expanded in decreasing odd powers of (2*x): U(2*n+1,x) = Sum_{m=0..n} T(n,m)*(2*x)^(2*(n-m)+1). See the W. Lang link given in A053125.
%C A053123 Unsigned version is mirror image of A078812. - _Philippe Deléham_, Dec 02 2008
%D A053123 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795
%D A053123 Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.
%D A053123 Stephen Barnett, "Matrices: Methods and Applications", Oxford University Press, 1990, p. 132, 343.
%H A053123 T. D. Noe, <a href="/A053123/b053123.txt">Rows n=0..50 of triangle, flattened</a>
%H A053123 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A053123 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A053123 T(n, m) = 0 if n<m else (-1)^m*binomial(2*n+1-m, m);
%F A053123 T(n, m) = -2*T(n-1, m-1) + T(n-1, m) - T(n-2, m-2), T(n, -2) = 0, T(-2, m) = 0, T(n, -1) = 0 = T(-1, m), T(0, 0) = 1, T(n, m) = 0 if n<m.
%F A053123 G.f. for m-th column (signed triangle): ((-1)^m)*x^m*Po(m+1, x)/(1-x)^(m+1), with Po(k, x) := Sum_{j=0..floor(k/2)} binomial(k, 2*j+1)*x^j.
%F A053123 The n-th degree polynomial is the characteristic equation for an n X n tridiagonal matrix with (diagonal = all 2's, sub and superdiagonals all -1's and the rest 0's), exemplified by the 4X4 matrix M = [2 -1 0 0 / -1 2 -1 0 / 0 -1 2 -1 / 0 0 -1 2]. - _Gary W. Adamson_, Jan 05 2005
%F A053123 Sum_{m=0..n} T(n,m)*(c(n))^(2*n-2*m) = 1/c(n), where c(n) = 2*cos(Pi/(2*n+3)). - _L. Edson Jeffery_, Sep 13 2013
%e A053123 Triangle begins:
%e A053123 1;
%e A053123 1, -2;
%e A053123 1, -4, 3;
%e A053123 1, -6, 10, -4;
%e A053123 1, -8, 21, -20, 5;
%e A053123 1, -10, 36, -56, 35, -6;
%e A053123 1, -12, 55, -120, 126, -56, 7; ...
%e A053123 E.g. fourth row (n=3) {1,-6,10,-4} corresponds to polynomial S(3,x-2) = x^3-6*x^2+10*x-4.
%p A053123 A053123 := proc(n,m)
%p A053123 (-1)^m*binomial(2*n+1-m,m) ;
%p A053123 end proc: # _R. J. Mathar_, Sep 08 2013
%t A053123 T[n_, m_]:= (-1)^m*Binomial[2*n+1-m, m]; Table[T[n, m], {n, 0, 11}, {m, 0, n}]//Flatten (* _Jean-François Alcover_, Mar 05 2014, after _R. J. Mathar_ *)
%o A053123 (PARI) for(n=0,10, for(k=0,n, print1((-1)^k*binomial(2*n-k+1,k), ", "))) \\ _G. C. Greubel_, Jul 23 2019
%o A053123 (Magma) [(-1)^k*Binomial(2*n-k+1,k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Jul 23 2019
%o A053123 (Sage) [[(-1)^k*binomial(2*n-k+1,k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Jul 23 2019
%o A053123 (GAP) Flat(List([0..10], n-> List([0..n], k-> (-1)^k*Binomial(2*n-k+1,k) ))); # _G. C. Greubel_, Jul 23 2019
%Y A053123 Cf. A053124, A053122, A172431.
%K A053123 easy,sign,tabl
%O A053123 0,3
%A A053123 _Wolfdieter Lang_