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A191897 Coefficients of the Z(n,x) polynomials; Z(0,x) = 1, Z(1,x) = x and Z(n,x) = x*Z(n-1,x) - 2*Z(n-2,x), n >= 2. 0
1, 1, 0, 1, 0, -2, 1, 0, -4, 0, 1, 0, -6, 0, 4, 1, 0, -8, 0, 12, 0, 1, 0, -10, 0, 24, 0, -8, 1, 0, -12, 0, 40, 0, -32, 0, 1, 0, -14, 0, 60, 0, -80, 0, 16, 1, 0, -16, 0, 84, 0, -160, 0, 80, 0, 1, 0, -18, 0, 112, 0, -280, 0, 240, 0, -32 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

The coefficients of the Z(n,x) polynomials by decreasing exponents, see the formulas, define this triangle.

LINKS

Table of n, a(n) for n=0..65.

FORMULA

Z(0,x) = 1, Z(1,x) = x and Z(n,x) = x*Z(n-1,x) - 2*Z(n-2,x), n >= 2.

a(n,k) = A077957(k) * A053119(n,k). - Paul Curtz, Sep 30 2011

EXAMPLE

The first few rows of the coefficients of the Z(n,x) are

  1;

  1,    0;

  1,    0,   -2;

  1,    0,   -4,    0;

  1,    0,   -6,    0,    4;

  1,    0,   -8,    0,   12,    0;

  1,    0,  -10,    0,   24,    0,   -8;

  1,    0,  -12,    0,   40,    0,  -32,    0;

  1,    0,  -14,    0,   60,    0,  -80,    0,   16;

  1,    0,  -16,    0,   84,    0, -160,    0,   80,    0;

MAPLE

nmax:=10: Z(0, x):=1 : Z(1, x):=x: for n from 2 to nmax do Z(n, x) := x*Z(n-1, x) - 2*Z(n-2, x) od: for n from 0 to nmax do for k from 0 to n do T(n, k) := coeff(Z(n, x), x, n-k) od: od: seq(seq(T(n, k), k=0..n), n=0..nmax); # Johannes W. Meijer, Jun 27 2011, revised Nov 29 2012

MATHEMATICA

a[n_, k_] := If[OddQ[k], 0, 2^(k/2)*Coefficient[ ChebyshevU[n, x/2], x, n-k]]; Flatten[ Table[ a[n, k], {n, 0, 10}, {k, 0, n}]] (* Jean-Fran├žois Alcover, Aug 02 2012, from 2nd formula *)

CROSSREFS

Row sums: A107920(n+1). Main diagonal: A077966(n).

Z(n,x=1) = A107920(n+1), Z(n,x=2)  = A009545(n+1),

Z(n,x=3) = A000225(n+1), Z(n,x=4)  = A007070(n),

Z(n,x=5) = A107839(n),   Z(n,x=6)  = A154244(n),

Z(n,x=7) = A186446(n),   Z(n,x=8)  = A190975(n+1),

Z(n,x=9) = A190979(n+1), Z(n,x=10) = A190869(n+1).

Row sum without sign: A113405(n+1).

Cf. A128099, A013609.

Sequence in context: A161552 A095859 A300482 * A088850 A185964 A143424

Adjacent sequences:  A191894 A191895 A191896 * A191898 A191899 A191900

KEYWORD

sign,tabl

AUTHOR

Paul Curtz, Jun 19 2011

EXTENSIONS

Edited and information added by Johannes W. Meijer, Jun 27 2011

STATUS

approved

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Last modified May 20 03:10 EDT 2019. Contains 323412 sequences. (Running on oeis4.)