A142475 Triangle T(n, k) = coefficients of (1 + x)/(1 + x + x^(k+2)), read by rows.

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.

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

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: A284606 A284019 A286135 * A051556 A330166 A081602

Adjacent sequences: A142472 A142473 A142474 * A142476 A142477 A142478

Keyword

sign,tabl,easy

Author

Roger L. Bagula, Sep 21 2008

Extensions

Edited by G. C. Greubel, Apr 13 2021

Status

approved