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A192174 Triangle T(n,k) of the coefficients [x^(n-k)] of the polynomial p(0,x)=-1, p(1,x)=x and p(n,x) = x*p(n-1,x) - p(n-2,x) in row n, column k.

%I A192174 #32 Mar 17 2019 03:47:19
%S A192174 -1,1,0,1,0,1,1,0,0,0,1,0,-1,0,-1,1,0,-2,0,-1,0,1,0,-3,0,0,0,1,1,0,-4,
%T A192174 0,2,0,2,0,1,0,-5,0,5,0,2,0,-1,1,0,-6,0,9,0,0,0,-3,0,1,0,-7,0,14,0,-5,
%U A192174 0,-5,0,1,1,0,-8,0,20,0,-14,0,-5,0,4,0
%N A192174 Triangle T(n,k) of the coefficients [x^(n-k)] of the polynomial p(0,x)=-1, p(1,x)=x and p(n,x) = x*p(n-1,x) - p(n-2,x) in row n, column k.
%C A192174 Consider the Catalan triangle A009766 antisymmetrically extended by a mirror along the diagonal (see also A176239):
%C A192174 0, -1, -1, -1, -1,  -1,  -1,   -1,
%C A192174 1,  0, -1, -2, -3,  -4,  -5,   -6,
%C A192174 1,  1,  0, -2, -5,  -9, -14,  -20,
%C A192174 1,  2,  2,  0, -5, -14, -28,  -48,
%C A192174 1,  3,  5,  5,  0, -14, -42,  -90,
%C A192174 1,  4,  9, 14, 14,   0, -42, -132,
%C A192174 1,  5, 14, 28, 42,  42,   0, -132,
%C A192174 1,  6, 20, 48, 90, 132, 132,    0.
%C A192174 The rows in this array are essentially the columns of T(n,k).
%F A192174 Sum_{k=0..n} T(n,k) = A057079(n-1).
%F A192174 Apparently T(3s,2s-2) = (-1)^(s+1)*A000245(s), s >= 1.
%e A192174 Triangle begins
%e A192174   -1;      # -1
%e A192174    1,  0;      # x
%e A192174    1,  0,  1;      # x^2+1
%e A192174    1,  0,  0,  0;      # x^3
%e A192174    1,  0, -1,  0, -1;      # x^4-x^2-1
%e A192174    1,  0, -2,  0, -1,  0;
%e A192174    1,  0, -3,  0,  0,  0,  1;
%e A192174    1,  0, -4,  0,  2,  0,  2,  0;
%e A192174    1,  0, -5,  0,  5,  0,  2,  0, -1;
%e A192174    1,  0, -6,  0,  9,  0,  0,  0, -3,  0;
%e A192174    1,  0, -7,  0, 14,  0, -5,  0, -5,  0,  1;
%e A192174    1,  0, -8,  0, 20,  0,-14,  0, -5,  0,  4,  0;
%e A192174    1,  0, -9,  0, 27,  0,-28,  0,  0,  0,  9,  0, -1;
%p A192174 p:= proc(n,x) option remember: if n=0 then -1 elif n=1 then x elif n>=2 then x*procname(n-1,x)-procname(n-2,x) fi: end: A192174 := proc(n,k): coeff(p(n,x),x,n-k): end: seq(seq(A192174(n,k),k=0..n), n=0..11); # _Johannes W. Meijer_, Aug 21 2011
%Y A192174 Cf. A023443, A080956 and A000096, A129936 and A005586, A005587.
%Y A192174 Cf. A194084. - _Paul Curtz_, Aug 16 2011
%K A192174 sign,tabl,easy
%O A192174 0,18
%A A192174 _Paul Curtz_, Jun 24 2011