A394670 Triangle read by rows: T(n,k) is the number of ways n students take an exam one by one on any number of consecutive days as long as there are at least k students for each day.

1, 0, 1, 0, 4, 2, 0, 24, 6, 6, 0, 192, 48, 24, 24, 0, 1920, 360, 120, 120, 120, 0, 23040, 3600, 1440, 720, 720, 720, 0, 322560, 40320, 15120, 5040, 5040, 5040, 5040, 0, 5160960, 524160, 161280, 80640, 40320, 40320, 40320, 40320, 0, 92897280, 7620480, 2177280, 1088640, 362880, 362880, 362880, 362880, 362880

Offset

0,5

References

Miklos Bona, Introduction to Enumerative and Analytic Combinatorics, CRC Press, 2025, pp. 159-160.

Links

Stefano Spezia, Table of n, a(n) for n = 0..5049 (first 100 rows of the triangle, flattened)

Formula

T(n,k) = n! * [x^n] 1/(1 - x^k/(1 - x)) with 0 <= k <= n.

Example

The triangle begins as:
  1;
  0,     1;
  0,     4,    2;
  0,    24,    6,    6;
  0,   192,   48,   24,  24;
  0,  1920,  360,  120, 120, 120;
  0, 23040, 3600, 1440, 720, 720, 720;
  ...

Mathematica

T[n_, k_]:=n!SeriesCoefficient[1/(1-x^k/(1-x)), {x, 0, n}]; Table[T[n, k], {n, 0, 9}, {k, 0, n}]//Flatten

Crossrefs

Columns k=0..3 give A000007, A002866, A052554, A052557.

The right diagonal gives A000142.

Row sums give A394677.

Sequence in context: A334778 A111549 A279411 * A022696 A371076 A019155

Adjacent sequences: A394667 A394668 A394669 * A394671 A394673 A394674

Keyword

nonn,easy,tabl

Author

Stefano Spezia, Mar 28 2026

Status

approved