0,2

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.

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.

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

Index entries for sequences related to Chebyshev polynomials.

W. Lang, First rows and related triangles .

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)).

{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.

Cf. A053124, A053123.

Sequence in context: A090881 A110485 A022664 this_sequence A038232 A084623 A137393

Adjacent sequences: A053122 A053123 A053124 this_sequence A053126 A053127 A053128

easy,sign,tabl

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)