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

0,2

Links

G. C. Greubel, Rows n = 0..30 of triangle, flattened

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

Example

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;

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

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 *)

Prog

(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

Crossrefs

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.

Sequence in context: A238660 A174793 A245718 * A152855 A218907 A192575

Adjacent sequences: A192008 A192009 A192010 * A192012 A192013 A192014

Keyword

sign,easy,tabl

Author

Paul Curtz, Jun 21 2011

Status

approved