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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.

%I A191897 #26 Feb 24 2019 01:58:28
%S A191897 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,
%T A191897 -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,
%U A191897 1,0,-18,0,112,0,-280,0,240,0,-32
%N 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.
%C A191897 The coefficients of the Z(n,x) polynomials by decreasing exponents, see the formulas, define this triangle.
%F A191897 Z(0,x) = 1, Z(1,x) = x and Z(n,x) = x*Z(n-1,x) - 2*Z(n-2,x), n >= 2.
%F A191897 a(n,k) = A077957(k) * A053119(n,k). - _Paul Curtz_, Sep 30 2011
%e A191897 The first few rows of the coefficients of the Z(n,x) are
%e A191897   1;
%e A191897   1,    0;
%e A191897   1,    0,   -2;
%e A191897   1,    0,   -4,    0;
%e A191897   1,    0,   -6,    0,    4;
%e A191897   1,    0,   -8,    0,   12,    0;
%e A191897   1,    0,  -10,    0,   24,    0,   -8;
%e A191897   1,    0,  -12,    0,   40,    0,  -32,    0;
%e A191897   1,    0,  -14,    0,   60,    0,  -80,    0,   16;
%e A191897   1,    0,  -16,    0,   84,    0, -160,    0,   80,    0;
%p A191897 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
%t A191897 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 *)
%Y A191897 Row sums: A107920(n+1). Main diagonal: A077966(n).
%Y A191897 Z(n,x=1) = A107920(n+1), Z(n,x=2)  = A009545(n+1),
%Y A191897 Z(n,x=3) = A000225(n+1), Z(n,x=4)  = A007070(n),
%Y A191897 Z(n,x=5) = A107839(n),   Z(n,x=6)  = A154244(n),
%Y A191897 Z(n,x=7) = A186446(n),   Z(n,x=8)  = A190975(n+1),
%Y A191897 Z(n,x=9) = A190979(n+1), Z(n,x=10) = A190869(n+1).
%Y A191897 Row sum without sign: A113405(n+1).
%Y A191897 Cf. A128099, A013609.
%K A191897 sign,tabl
%O A191897 0,6
%A A191897 _Paul Curtz_, Jun 19 2011
%E A191897 Edited and information added by _Johannes W. Meijer_, Jun 27 2011