Richard L. Ollerton - cp's OEIS frontend

User: Richard L. Ollerton

Richard L. Ollerton has authored 2 sequences.

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

Richard L. Ollerton, Nov 11 2021

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

			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).

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

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

Cf. A007318, A051731, A054521.

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

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

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.

A039671 Row sums up to the main diagonal of the "postage stamp" array (n,m >= 0) defined in A007059.

Original entry on oeis.org

1, 1, 3, 8, 21, 53, 130, 310, 724, 1661, 3757, 8398, 18588, 40800, 88918, 192592, 414907, 889631, 1899554, 4040864, 8567342, 18109698, 38176280, 80278798, 168432854, 352658013, 736977583, 1537420460, 3202035086, 6658948608

Offset: 0

Richard L. Ollerton and Eirwyn L. Ollerton

			a(0)=1, a(1)=0+1=1, a(2)=0+1+2=3, a(3)=0+1+3+4=8.

Cf. A007059.

f[ n_, m_ ] := f[ n, m ]=Which[ n>0, Sum[ f[ n-i, m ], {i, 1, m} ], n<0, 0, n==0, 1 ] Table[ Sum[ f[ n, j ], {j, 0, n} ], {n, 0, 30} ]

Conjecture: a(n) = Sum_{k=1..n} A126198(n,k), for n > 0. - L. Edson Jeffery, Nov 29 2013

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User: Richard L. Ollerton

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).

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