A390469 Array read by ascending antidiagonals: A(n,k) = [x^n] (1 - x^k)/(1 - 2*x + x^(1+k)), with k >= 0.

0, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 2, 1, 0, 1, 5, 4, 2, 1, 0, 1, 8, 7, 4, 2, 1, 0, 1, 13, 13, 8, 4, 2, 1, 0, 1, 21, 24, 15, 8, 4, 2, 1, 0, 1, 34, 44, 29, 16, 8, 4, 2, 1, 0, 1, 55, 81, 56, 31, 16, 8, 4, 2, 1, 0, 1, 89, 149, 108, 61, 32, 16, 8, 4, 2, 1, 0, 1, 144, 274, 208, 120, 63, 32, 16, 8, 4, 2, 1

Offset

0,9

Comments

A(n,k) is the number of possible outcome sequences of length n of fair coin flips that do not contain k consecutive heads.

References

Miklos Bona, Introduction to Enumerative and Analytic Combinatorics, CRC Press, 2025, pp. 391-392.

Links

Table of n, a(n) for n=0..90.

Example

The array begins as:
  0, 1,  1,  1,   1,   1,   1, ...
  0, 1,  2,  2,   2,   2,   2, ...
  0, 1,  3,  4,   4,   4,   4, ...
  0, 1,  5,  7,   8,   8,   8, ...
  0, 1,  8, 13,  15,  16,  16, ...
  0, 1, 13, 24,  29,  31,  32, ...
  0, 1, 21, 44,  56,  61,  63, ...
  0, 1, 34, 81, 108, 120, 125, ...
  ...

Mathematica

A[n_, k_]:=SeriesCoefficient[(1-x^k)/(1-2x+x^(1+k)), {x, 0, n}]; Table[A[n-k, k], {n, 0, 12}, {k, 0, n}]//Flatten

Crossrefs

Columns k=0..16 give: A000004, A000012, A000045, A000073, A000078, A001591, A001592, A066178, A079262, A104144, A122265, A168082, A168083, A168084, A220469, A220493, A249169.

Main diagonal gives A000225.

Rows n=0..2 give: A057427, A130130, A113311.

Antidiagonal sums give A048888.

Sequence in context: A332670 A118344 A343138 * A119270 A267109 A341524

Adjacent sequences: A390466 A390467 A390468 * A390470 A390471 A390472

Keyword

nonn,easy,tabl

Author

Stefano Spezia, Apr 05 2026

Status

approved