A349221 Triangle read by rows: T(n, k) = 1 if k divides binomial(n-1, k-1), T(n, k) = 0 otherwise (n >= 1, 1 <= k <= n).

1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0

Offset: 1

Author

Richard L. Ollerton, Nov 11 2021

Keywords

• nonn
• tabl
• look

Comments

Similar to A054521 as gcd(n, k) = 1 => k divides binomial(n-1, k-1) but not equivalent as the converse is not true, the earliest example being T(10,4) where 4 divides binomial(9,3) = 84 but gcd(10,4) is not 1. Question: What characterizes the cases where this triangle differs from A054521?

The period of the k-th column is given by A349593(k-1, k) = k * Product_{prime p|k} p^(floor(log(k-1)/log(p))). - Jianing Song, Nov 29 2021

{T(n, k)} is the sum of triangles [k|binomial(n-1, k-1) AND gcd(n, k) = j], n >= 1, 1 <= k <= n, j >= 1, where [] is the Iverson bracket. For j > 1, bitmaps of these triangles suggest simpler fractal gaskets that combine to produce the "shadowing" effect observed in the bitmap of {T(n, k)} provided in the LINKS section. For prime j, the bitmaps suggest a fractal (Hausdorff) dimension of log(A000217(j))/log(j) = log(j(j + 1)/2)/log(j), which is the same as that of the gasket formed by taking the Pascal triangle (A007318) mod j (see Bondarenko reference). - Richard L. Ollerton, Dec 10 2021

Examples

			The triangle T(n, k) begins:
n\k  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 ...
1:   1
2:   1  0
3:   1  1  0
4:   1  0  1  0
5:   1  1  1  1  0
6:   1  0  0  0  1  0
7:   1  1  1  1  1  1  0
8:   1  0  1  0  1  0  1  0
9:   1  1  0  1  1  0  1  1  0
10:  1  0  1  1  0  1  1  0  1  0
11:  1  1  1  1  1  1  1  1  1  1  0
12:  1  0  0  0  1  1  1  0  0  0  1  0
13:  1  1  1  1  1  1  1  1  1  1  1  1  0
14:  1  0  1  0  1  0  0  0  1  0  1  0  1  0
15:  1  1  0  1  0  0  1  1  0  0  1  0  1  1  0
...
Differences between this example and that for A054521 occur at (n,k) = (10,4), (10,6), and (12,6).
		

References

• Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1 <= n <= 150, flattened)
• Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see pp. 130-132.
• Michael De Vlieger, Bitmap of rows 1 <= n <= 2^10, showing 1 as black and 0 as white.
• Michael De Vlieger, Table of b(n) for n = 1..3322, where b(n) is the compactification of row n of a(n) as a binary number.
• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

Cf. A007318, A051731, A054521.

Programs

Mathematica

Table[Boole[Mod[Binomial[n - 1, k - 1], k] == 0], {n, 12}, {k, n}] // Flatten (* Michael De Vlieger, Nov 11 2021 *)

PARI

row(n) = vector(n, k, !(binomial(n-1,k-1) % k)); \\ Michel Marcus, Nov 11 2021

Formula

T(n, k) = [k|binomial(n-1, k-1)] = Sum_{j>=1} [k|binomial(n-1, k-1) AND gcd(n, k) = j], n >= 1, 1 <= k <= n, where [] is the Iverson bracket. (The j = 1 case is A054521.)

T(n, k) = T(n, n-k), n > 1, 1 <= k < n.