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Let P(0,x) = -1, P(1,x) = 2*x, and P(n,x) = x*P(n-1,x) - P(n-2,x) for n > 1. This sequence is the triangle of polynomial coefficients in order of decreasing exponents.
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%I #35 May 21 2019 03:49:00

%S -1,2,0,2,0,1,2,0,-1,0,2,0,-3,0,-1,2,0,-5,0,0,0,2,0,-7,0,3,0,1,2,0,-9,

%T 0,8,0,1,0,2,0,-11,0,15,0,-2,0,-1,2,0,-13,0,24,0,-10,0,-2,0,2,0,-15,0,

%U 35,0,-25,0,0,0,1,2,0,-17,0,48,0,-49,0,10,0,3,0,2,0,-19,0,63,0,-84,0,35,0,3,0,-1

%N Let P(0,x) = -1, P(1,x) = 2*x, and P(n,x) = x*P(n-1,x) - P(n-2,x) for n > 1. This sequence is the triangle of polynomial coefficients in order of decreasing exponents.

%H G. C. Greubel, <a href="/A192011/b192011.txt">Rows n = 0..30 of triangle, flattened</a>

%F T(n, k) = T(n-1, k) - T(n-2, k-2), where T(0, 0) = -1, T(n, 0) = 2 and 0 <= k <= n, n >= 0. - _G. C. Greubel_, May 19 2019

%e The first few rows are

%e -1;

%e 2, 0;

%e 2, 0, 1;

%e 2, 0, -1, 0;

%e 2, 0, -3, 0, -1;

%e 2, 0, -5, 0, 0, 0;

%e 2, 0, -7, 0, 3, 0, 1;

%e 2, 0, -9, 0, 8, 0, 1, 0;

%e 2, 0, -11, 0, 15, 0, -2, 0, -1;

%e 2, 0, -13, 0, 24, 0, -10, 0, -2, 0;

%e 2, 0, -15, 0, 35, 0, -25, 0, 0, 0, 1;

%p A192011 := proc(n,k)

%p option remember;

%p if k>n or k <0 or n<0 then

%p 0;

%p elif n= 0 then

%p -1;

%p elif k=0 then

%p 2;

%p else

%p procname(n-1,k)-procname(n-2,k-2) ;

%p end if;

%p end proc: # _R. J. Mathar_, Nov 03 2011

%t p[0, _] = -1; p[1, x_] := 2x; p[n_, x_] := p[n, x] = x*p[n-1, x] - p[n-2, x]; row[n_] := CoefficientList[p[n, x], x]; Table[row[n] // Reverse, {n, 0, 9}] // Flatten (* _Jean-François Alcover_, Nov 26 2012 *)

%t T[n_,k_]:= If[k<0 || k>n, 0, If[n==0 && k==0, -1, If[k==0, 2, T[n-1,k] - T[n-2, k-2]]]]; Table[T[n,k], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, May 19 2019 *)

%o (PARI) {T(n,k) = if(k<0 || k>n, 0, if(n==0 && k==0, -1, if(k==0, 2, T(n-1,k) - T(n-2,k-2)))) };

%o for(n=0, 10, for(k=0, n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, May 19 2019

%o (Sage)

%o def T(n,k):

%o if (k<0 or k>n): return 0

%o elif (n==0 and k==0): return -1

%o elif (k==0): return 2

%o else: return T(n-1,k) - T(n-2, k-2)

%o [[T(n,k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, May 19 2019

%Y Left hand diagonals are: T(n,0) = [-1,2,2,2,2,2,...], T(n,2) = A165747(n), T(n,4) = A067998(n+1), T(n,6) = -A058373(n), T(n,8) = (-1)^(n+1) * A167387(n+2) see also A052472(n.

%K sign,easy,tabl

%O 0,2

%A _Paul Curtz_, Jun 21 2011