A386791 Triangle read by rows: T(n, k) = (n - k)/k if k divides n - k else 0 for k > 0, T(n, 0) = 0^n.

1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 3, 1, 0, 0, 0, 4, 0, 0, 0, 0, 0, 5, 2, 1, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 7, 3, 0, 1, 0, 0, 0, 0, 0, 8, 0, 2, 0, 0, 0, 0, 0, 0, 0, 9, 4, 0, 0, 1, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 5, 3, 2, 0, 1, 0, 0, 0, 0, 0, 0

Offset

0,8

Links

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

Formula

Sum_{k=0..n} T(n, k) = A000203(n) - A000005(n) = A065608(n) for n > 0.

sign(T(n, k)) = A175992(n, k) for n, k >= 1.

T(2*n, n) = A000012(n).

Example

Triangle begins:
  [ 0] 1;
  [ 1] 0,  0;
  [ 2] 0,  1, 0;
  [ 3] 0,  2, 0, 0;
  [ 4] 0,  3, 1, 0, 0;
  [ 5] 0,  4, 0, 0, 0, 0;
  [ 6] 0,  5, 2, 1, 0, 0, 0;
  [ 7] 0,  6, 0, 0, 0, 0, 0, 0;
  [ 8] 0,  7, 3, 0, 1, 0, 0, 0, 0;
  [ 9] 0,  8, 0, 2, 0, 0, 0, 0, 0, 0;
  [10] 0,  9, 4, 0, 0, 1, 0, 0, 0, 0, 0;
  [11] 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
  [12] 0, 11, 5, 3, 2, 0, 1, 0, 0, 0, 0, 0, 0;

Maple

A386791 := (n, k) -> ifelse(k = 0, 0^n, ifelse(modp(n-k, k) = 0, iquo(n-k, k), 0)):
seq(seq(A386791(n, k), k = 0..n), n = 0..12);

Mathematica

A386791[n_, k_] := If[k == 0, Boole[k == n], If[Divisible[n-k, k], Quotient[n-k, k], 0]];
Table[A386791[n, k], {n, 0, 12}, {k, 0, n}] // Flatten

Prog

(SageMath)
def A386791(n, k):
    if k == 0: return int(k == n)
    return (n - k) // k if k.divides(n - k) else 0
for n in (0..12):
    print([A386791(n, k) for k in (0..n)])

Crossrefs

Cf. A000007 (column 0 and main diagonal), A065608 (row sums), A000012 (central terms), A175992 (subtriangle of sign), A386790.

Sequence in context: A347233 A327172 A355524 * A392491 A113503 A082507

Adjacent sequences: A386788 A386789 A386790 * A386792 A386793 A386794

Keyword

nonn,tabl

Author

Peter Luschny, Aug 04 2025

Status

approved