A053125 Triangle of coefficients of Chebyshev's U(n,2*x-1) polynomials (exponents of x in decreasing order).

1, 4, -2, 16, -16, 3, 64, -96, 40, -4, 256, -512, 336, -80, 5, 1024, -2560, 2304, -896, 140, -6, 4096, -12288, 14080, -7680, 2016, -224, 7, 16384, -57344, 79872, -56320, 21120, -4032, 336, -8, 65536, -262144, 430080, -372736, 183040, -50688, 7392, -480, 9, 262144, -1179648, 2228224, -2293760, 1397760

Offset

0,2

Comments

A000302 (powers of 4), A002699, A002700 unsigned column sequences for m=0..2.

G.f. for row polynomials U(n,2*x-1) and row sums same as for A053124.

With offset 1 this is also the coefficient triangle of 2* U(2*n-1,x) expanded in decreasing powers of x. W. Lang, Mar 07 2007.

References

C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 518.

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

Links

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

W. Lang, First rows and related triangles .

Eric Weisstein's World of Mathematics, Chebyshev Polynomial of the Second Kind

Index entries for sequences related to Chebyshev polynomials.

Formula

a(n, m) = A053124(n, n-m)= (4^(n-m))*A053123(n, m)= (4^(n-m))*((-1)^m)*binomial(2*n+1-m, m) if n >= m, else 0.

a(n, m) := -2*a(n-1, m-1)+4*a(n-1, m)-a(n-2, m-2), a(-2, m) := 0=: a(n, -2), a(-1, m) := 0=: a(n, -1), a(0, 0)=1, a(n, m)=0 if n<m;

G.f. for m-th column (signed triangle): ((-x)^m)*Po(m+1, 4*x)/(1-4*x)^(m+1), with Po(k, x) := sum('binomial(k, 2*j+1)*x^j', 'j'=0..floor(k/2)).

Example

{1}; {4,-2}; {16,-16,3}; {64,-96,40,-4}; {256,-512,336,-80,5};... E.g. fourth row (n=3) corresponds to polynomial U^{*}(3,m)=U(3,2*x-1)= 64*x^3-96*x^2+40*x-4.

Mathematica

Reverse /@ CoefficientList[Table[ChebyshevU[n, 2 x - 1], {n, 0, 10}], x] // Flatten (* Eric W. Weisstein, Apr 04 2018 *)
Reverse /@ CoefficientList[ChebyshevU[Range[0, 10], 2 x - 1], x] // Flatten (* Eric W. Weisstein, Apr 04 2018 *)

Crossrefs

Cf. A053124, A053123.

Sequence in context: A022664 A316463 A167784 * A038232 A254632 A084623

Adjacent sequences: A053122 A053123 A053124 * A053126 A053127 A053128

Keyword

easy,sign,tabl

Author

Wolfdieter Lang

Status

approved