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A264872 := proc(n, m)

2^n*(1+2^n)^m ;

end proc:

seq(seq(A264872(n, d-n), n=0..d), d=0..12) ; # R. J. Mathar, Aug 14 2024

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R. J. Mathar, <a href="http://vixra.org/abs/1511.0225">Counting 2-way monotonic terrace forms over rectangular landscapes</a>, vixra:1511.0225 (2015), Section 6.

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Start with a an n X m rectangle and cut it vertically along any set of the m-1 separators. There are binomial(m-1,c) ways of doing this with 0 <= c < m cuts. Inside each of these 1+c regions cut vertically, of for which there are 2^(n-1) choices. The total number of ways of dissecting the rectangle in into rectangles in this way is Sum_{c=0..m-1} binomial(m-1,c) 2^((1+c)(n-1)) = 2^(n-1)*(1+2^(n-1))^(m-1) = T(n-1,m-1).

The symmetrized version of the array is S(n,m) = T(n,m) + T(m,n) - 2^(m+n) <= A116694(n,m), which counts tilings which that start with Guillotine guillotine cuts either horizontally or vertically, avoiding double account for counting of the tilings where the order of the cuts does not matter. - R. J. Mathar, Nov 29 2015

.

1, 2, 4, 8, 16, 32, ...

2, 8, 30, 110, 402, 1478, ...

4, 30, 184, 1116, 7060, 47220, ...

8, 110, 1116, 11600, 130968, 1622120, ...

16, 402, 7060, 130968, 2672416, 60666672, ...

32, 1478, 47220, 1622120, 60666672, 2504664128, ...

Cf. A000079 (row and column 0), A008776 (row 1), A005054 (row 2), A055275 (row 3), A063376 (column 1),.

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The symmetrized version of the array is S(n,m) = T(n,m)+T(m,n)-2^(m+n) <= A116694(n,m), which counts tilings which start with Guillotine cuts either horizontally or vertically, avoiding double account for the tilings where the order of the cuts does not matter. - R. J. Mathar, Nov 29 2015

The symmetrized version S(n,m) starts

1, 2, 4, 8, 16, 32,

2, 8, 30, 110, 402, 1478,

4, 30, 184, 1116, 7060, 47220,

8, 110, 1116, 11600, 130968, 1622120,

16, 402, 7060, 130968, 2672416,60666672,

32, 1478, 47220, 1622120,60666672,2504664128,

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