A192011 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.
-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, 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, 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
Offset: 0
Examples
The first few rows are -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, 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, 35, 0, -25, 0, 0, 0, 1;
Links
- G. C. Greubel, Rows n = 0..30 of triangle, flattened
Crossrefs
Programs
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Maple
A192011 := proc(n,k) option remember; if k>n or k <0 or n<0 then 0; elif n= 0 then -1; elif k=0 then 2; else procname(n-1,k)-procname(n-2,k-2) ; end if; end proc: # R. J. Mathar, Nov 03 2011
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Mathematica
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[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 *)
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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)))) }; for(n=0, 10, for(k=0, n, print1(T(n,k), ", "))) \\ G. C. Greubel, May 19 2019 -
Sage
def T(n,k): if (k<0 or k>n): return 0 elif (n==0 and k==0): return -1 elif (k==0): return 2 else: return T(n-1,k) - T(n-2, k-2) [[T(n,k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 19 2019
Formula
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