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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. 9
-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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified July 10 00:00 EDT 2020. Contains 335570 sequences. (Running on oeis4.)