A358125 -id:A358125 - cp's OEIS frontend

A359200 Triangle read by rows: T(n, k) = A358125(n,k)*binomial(n-1, k), 0 <= k <= n-1.

0, 1, 1, 3, 8, 3, 7, 30, 30, 7, 15, 88, 144, 88, 15, 31, 230, 520, 520, 230, 31, 63, 564, 1620, 2240, 1620, 564, 63, 127, 1330, 4620, 8120, 8120, 4620, 1330, 127, 255, 3056, 12432, 26432, 33600, 26432, 12432, 3056, 255, 511, 6894, 32112, 79968, 122976, 122976, 79968, 32112, 6894, 511

Offset: 1

Ambrosio Valencia-Romero, Dec 20 2022

			Triangle begins:
   0;
   1,     1;
   3,     8,     3;
   7,    30,    30,      7;
  15,    88,   144,     88,     15;
  31,   230,   520,    520,    230,     31;
  63,   564,  1620,   2240,   1620,    564,     63;
 127,  1330,  4620,   8120,   8120,   4620,   1330,    127;
 255,  3056, 12432,  26432,  33600,  26432,  12432,   3056,   255;
 511,  6894, 32112,  79968, 122976, 122976,  79968,  32112,  6894,   511;
1023, 15340, 80460, 229440, 413280, 499968, 413280, 229440, 80460, 15340, 1023;
...

Row sums give 2*A005061(n-1).

T := n -> local k; seq((2^n - 2^(n - k - 1) - 2^k)*binomial(n - 1, k), k = 0 .. n - 1);
seq(T(n), n = 1 .. 11);

T[n_, k_] := (2^n - 2^(n - k - 1) - 2^k)*Binomial[n - 1, k]; Table[T[n, k], {n, 1, 10}, {k, 0, n - 1}] // Flatten (* Amiram Eldar, Dec 20 2022 *)

T(n, k) = (2^n - 2^(n-k-1) - 2^k)*binomial(n-1, k), for n >= 1 and 0 <= k <= n-1.

A357298 Triangle read by rows where all entries in every even row are 1's and the entries in every odd row alternate between 0 (start/end) and 1.

Original entry on oeis.org

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

Offset: 1

Ambrosio Valencia-Romero, Dec 20 2022

Row sums are equal to n for even n and (n-1)/2 for odd n; or A065423(n+1).

			Triangle begins:
   n\k  0  1  2  3  4  5  6  7  8  9 ...
   1    0;
   2    1, 1;
   3    0, 1, 0;
   4    1, 1, 1, 1;
   5    0, 1, 0, 1, 0;
   6    1, 1, 1, 1, 1, 1;
   7    0, 1, 0, 1, 0, 1, 0;
   8    1, 1, 1, 1, 1, 1, 1, 1;
   9    0, 1, 0, 1, 0, 1, 0, 1, 0;
  10    1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
  ...
Formatted as a symmetric triangle -- regular hexagram pattern with 0's at the centers formed by connecting all 1's:
      .----------------------------------------------.
      |                      k=0   1   2   3   4   5 |
      |-----------------------/---/---/---/---/--./  |
-------                          /   /   /   /   /   |
| n=1 |                     0   /   /   /   /   /|   |
-------                            /   /   /   / | 6 |
|   2 |                   1---1   /   /   /   /  |/  |
-------                    \ /       /   /   /   /   |
|   3 |                 0   1   0   /   /   /   /|   |
-------                    / \         /   /   / | 7 |
|   4 |               1---1---1---1   /   /   /  |/  |
-------                \ /     \ /       /   /   /   |
|   5 |             0   1   0   1   0   /   /   /|   |
-------                / \     / \         /   / | 8 |
|   6 |           1---1---1---1---1---1   /   /  |/  |
-------            \ /     \ /     \ /       /   /   |
|   7 |         0   1   0   1   0   1   0   /   /|   |
-------            / \     / \     / \         / | 9 |
|   8 |       1---1---1---1---1---1---1---1   /   /  |
-------        \ /     \ /     \ /     \ /       /   |
|   9 |     0   1   0   1   0   1   0   1   0   /|   |
-------        / \     / \     / \     / \       | . |
|  10 |   1---1---1---1---1---1---1---1---1---1  | . |
-------                                          | . |

Cf. A358125, A065423 (row sums).

T := n -> local k; seq(1/2 + 1/2*(-1)^(n*(k + 1)), k = 0 .. n - 1); # formula 1
seq(T(n), n=1..16); # print first 16 rows of formula 1.

T(n,k) = bitnegimply(1,n) || bitand(1,k); \\ Kevin Ryde, Dec 21 2022

T(n, k) = 1/2 + (1/2)*(-1)^(n*(k+1)), for n >= 1 and 0 <= k <= n-1.
T(n, k) = (2^n - 2^(n-k-1) - 2^k) mod 3, for n >= 1 and 0 <= k <= n-1.
T(n, k) = A358125(n, k) mod 3, for n >= 1 and 0 <= k <= n-1.

cp's OEIS Frontend

A359200 Triangle read by rows: T(n, k) = A358125(n,k)*binomial(n-1, k), 0 <= k <= n-1.

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