A386877 Triangle read by rows: T(n, k) = n! / (k! * (n/k)!) if k divides n otherwise 0; T(n, 0) = 0^n.

1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 6, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 60, 60, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 840, 0, 840, 0, 0, 0, 1, 0, 1, 0, 10080, 0, 0, 0, 0, 0, 1, 0, 1, 15120, 0, 0, 15120, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Author

Peter Bala and Peter Luschny, Aug 09 2025

Keywords

• nonn
• tabl

Examples

			Triangle starts:
  [ 0] [1]
  [ 1] [0, 1]
  [ 2] [0, 1,     1]
  [ 3] [0, 1,     0,     1]
  [ 4] [0, 1,     6,     0,   1]
  [ 5] [0, 1,     0,     0,   0,     1]
  [ 6] [0, 1,    60,    60,   0,     0, 1]
  [ 7] [0, 1,     0,     0,   0,     0, 0, 1]
  [ 8] [0, 1,   840,     0, 840,     0, 0, 0, 1]
  [ 9] [0, 1,     0, 10080,   0,     0, 0, 0, 0, 1]
  [10] [0, 1, 15120,     0,   0, 15120, 0, 0, 0, 0, 1]
  [11] [0, 1,     0,     0,   0,     0, 0, 0, 0, 0, 0, 1]
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).

Crossrefs

Cf. A121860 (row sums), A113704 (sign).

Programs

Mathematica

A386877[n_, k_] := Which[k == 0, Boole[n == 0], Divisible[n, k], n!/(k!*(n/k)!), True, 0];
Table[A386877[n, k], {n, 0, 12}, {k, 0, n}] (* Paolo Xausa, Aug 09 2025 *)

SageMath

F = factorial
def T(n, k):
    if k == 0: return 0**n
    return F(n)/(F(k)*F(n//k)) if k.divides(n) else 0
for n in range(33): print([T(n,k) for k in srange(n+1)])

Formula

sign(T(n, k)) = A113704(n, k).