A359574 Array read by antidiagonals: T(m,n) is the number of m X n binary arrays with all 1's connected and a path of 1's from top row to bottom row.

1, 3, 1, 6, 7, 1, 10, 28, 17, 1, 15, 88, 144, 41, 1, 21, 245, 920, 730, 99, 1, 28, 639, 5191, 9362, 3692, 239, 1, 36, 1608, 27651, 104989, 94280, 18666, 577, 1, 45, 3968, 143342, 1111283, 2075271, 947760, 94384, 1393, 1, 55, 9689, 733512, 11457514, 42972329, 40792921, 9528128, 477264, 3363, 1

Offset: 1

Author

Andrew Howroyd, Jan 06 2023

Keywords

• nonn
• tabl

Comments

The grid has m rows and n columns.

Examples

			Array begins:
================================================================
m\n| 1   2     3       4         5           6             7
---+------------------------------------------------------------
1  | 1   3     6      10        15          21            28 ...
2  | 1   7    28      88       245         639          1608 ...
3  | 1  17   144     920      5191       27651        143342 ...
4  | 1  41   730    9362    104989     1111283      11457514 ...
5  | 1  99  3692   94280   2075271    42972329     866126030 ...
6  | 1 239 18666  947760  40792921  1642690309   64270256276 ...
7  | 1 577 94384 9528128 801218515 62618577481 4741764527414 ...
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..435
• Chaim Goodman-Strauss, Notes on the number of m × n binary arrays with all 1's connected and a path of 1's from top row to bottom row (May 21 2020)

Crossrefs

Main diagonal is A163028.

Rows 1..10 are A000217, A163037, A163038, A163039, A163040, A163041, A163042, A163043, A163044, A163045.

Columns 1..10 are A000012, A001333(n+1), A163029, A163030, A163031, A163032, A163033, A163034, A163035, A163036.

Cf. A287151, A292357, A359573, A359576.

Formula

T(m,n) = A287151(m,n) - 2*A287151(m-1,n) + A287151(m-2,n) for m > 2.