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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
OFFSET
0,6
COMMENTS
The coefficients of the Z(n,x) polynomials by decreasing exponents, see the formulas, define this triangle.
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-François 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).
Sequence in context: A095859 A349627 A300482 * A088850 A363032 A185964
KEYWORD
sign,tabl
AUTHOR
Paul Curtz, Jun 19 2011
EXTENSIONS
Edited and information added by Johannes W. Meijer, Jun 27 2011
STATUS
approved