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A142475
Triangle T(n, k) = coefficients of (1 + x)/(1 + x + x^(k+2)), read by rows.
1
1, 0, 0, -1, 0, 0, 1, -1, 0, 0, 0, 1, -1, 0, 0, -1, -1, 1, -1, 0, 0, 1, 2, -1, 1, -1, 0, 0, 0, -3, 1, -1, 1, -1, 0, 0, -1, 4, 0, 1, -1, 1, -1, 0, 0, 1, -6, -1, -1, 1, -1, 1, -1, 0, 0, 0, 9, 2, 2, -1, 1, -1, 1, -1, 0, 0, -1, -13, -3, -3, 1, -1, 1, -1, 1, -1, 0, 0, 1, 19, 3, 4, 0, 1, -1, 1, -1, 1, -1, 0, 0, 0, -28, -2, -5, -1, -1, 1, -1, 1, -1, 1, -1, 0, 0
OFFSET
0,23
REFERENCES
Taylor L. Booth, Sequential Machines and Automata Theory, John Wiley and Sons, Inc., 1967, page 331ff.
FORMULA
T(n, k) = coefficients of (1 + x)/(1 + x + x^(k+2)).
EXAMPLE
Triangle begins as:
1;
0, 0;
-1, 0, 0;
1, -1, 0, 0;
0, 1, -1, 0, 0;
-1, -1, 1, -1, 0, 0;
1, 2, -1, 1, -1, 0, 0;
0, -3, 1, -1, 1, -1, 0, 0;
-1, 4, 0, 1, -1, 1, -1, 0, 0;
1, -6, -1, -1, 1, -1, 1, -1, 0, 0;
0, 9, 2, 2, -1, 1, -1, 1, -1, 0, 0;
-1, -13, -3, -3, 1, -1, 1, -1, 1, -1, 0, 0;
1, 19, 3, 4, 0, 1, -1, 1, -1, 1, -1, 0, 0;
0, -28, -2, -5, -1, -1, 1, -1, 1, -1, 1, -1, 0, 0;
-1, 41, 0, 6, 2, 2, -1, 1, -1, 1, -1, 1, -1, 0, 0;
MATHEMATICA
T[n_, k_]:= T[n, k]= SeriesCoefficient[Series[(1+t)/(1+t+t^(k+2)), {t, 0, n}], n];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Apr 13 2021 *)
PROG
(Sage)
def T(n, k): return ( (1+x)/(1+x+x^(k+2)) ).series(x, n+1).list()[n]
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 13 2021
CROSSREFS
Cf. A078012.
Sequence in context: A284019 A286135 A375010 * A051556 A330166 A081602
KEYWORD
sign,tabl,easy
AUTHOR
Roger L. Bagula, Sep 21 2008
EXTENSIONS
Edited by G. C. Greubel, Apr 13 2021
STATUS
approved