A053123 Triangle of coefficients of shifted Chebyshev's S(n,x-2)= U(n,x/2-1) polynomials (exponents of x in decreasing order).

1, 1, -2, 1, -4, 3, 1, -6, 10, -4, 1, -8, 21, -20, 5, 1, -10, 36, -56, 35, -6, 1, -12, 55, -120, 126, -56, 7, 1, -14, 78, -220, 330, -252, 84, -8, 1, -16, 105, -364, 715, -792, 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

Offset

0,3

Comments

T(n,m)= A053122(n,n-m).

G.f. for row polynomials and row sums same as in A053122.

Unsigned column sequences are A000012, A005843, A014105, A002492 for m=0..3, resp. and A053126-A053131 for m=4..9.

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.

Unsigned version is mirror image of A078812. - Philippe Deléham, Dec 02 2008

References

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

Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

Stephen Barnett, "Matrices: Methods and Applications", Oxford University Press, 1990, p. 132, 343.

Links

T. D. Noe, Rows n=0..50 of triangle, flattened

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Index entries for sequences related to Chebyshev polynomials.

Formula

T(n, m) = 0 if n<m else (-1)^m*binomial(2*n+1-m, m);

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.

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('binomial(k, 2*j+1)*x^j', 'j'=0..floor(k/2)).

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

sum(m=0..n, a(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

Example

Triangle begins:
  1;
  1,  -2;
  1,  -4,  3;
  1,  -6, 10,   -4;
  1,  -8, 21,  -20,   5;
  1, -10, 36,  -56,  35,  -6;
  1, -12, 55, -120, 126, -56, 7; ...
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.

Maple

A053123 := proc(n, m)
    (-1)^m*binomial(2*n+1-m, m) ;
end proc: # R. J. Mathar, Sep 08 2013

Mathematica

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 *)

Prog

(PARI) for(n=0, 10, for(k=0, n, print1((-1)^k*binomial(2*n-k+1, k), ", "))) \\ G. C. Greubel, Jul 23 2019
(Magma) [(-1)^k*Binomial(2*n-k+1, k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jul 23 2019
(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
(GAP) Flat(List([0..10], n-> List([0..n], k-> (-1)^k*Binomial(2*n-k+1, k) ))); # G. C. Greubel, Jul 23 2019

Crossrefs

Cf. A053124, A053122, A172431.

Sequence in context: A132191 A094437 A172431 * A107661 A126570 A048790

Adjacent sequences: A053120 A053121 A053122 * A053124 A053125 A053126

Keyword

easy,sign,tabl

Author

Wolfdieter Lang

Status

approved