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A264841 Triangle read by rows: T(n,k) is the number of ways to partition an n X k square grid into any number of parts along the gridlines. 1
1, 2, 12, 4, 74, 1442, 8, 456, 28028, 1716098, 16, 2810, 544844, 105093828, 20276816980, 32, 17316, 10591310, 6435880414, 3912156203494, 2378025136264102, 64, 106706, 205886234, 394129505248, 754801786191820, 1445496758320387318, 2768227968406304217000, 128, 657552, 4002256640, 24136256828880 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A set of edges forms a valid partition if and only if it includes the entire boundary of the grid, and there are no vertices of degree 1.

LINKS

Table of n, a(n) for n=1..32.

Danny Rorabaugh, A264841 Example: T(2,2)

R. J. Mathar, Counting 2-way monotonic terrace forms over rectangular landscapes, see Section 6.3, Combinatorics and Graph Theory, viXra:1511.0225, 2015.

FORMULA

T(n,1) = 2^(n-1).

T(n,2) = A078469(n).

EXAMPLE

The triangle T(n,k) begins:

n\k 1  2    3      4         5

1:  1

2:  2  12

3:  4  74   1442

4:  8  456  28028  1716098

5:  16 2810 544844 105093828 20276816980

CROSSREFS

A078469 is the second column of this triangle.

Sequence in context: A066700 A159323 A038218 * A191249 A333544 A005760

Adjacent sequences:  A264838 A264839 A264840 * A264842 A264843 A264844

KEYWORD

nonn,tabl

AUTHOR

Linus Hamilton, Nov 26 2015

STATUS

approved

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Last modified October 21 23:34 EDT 2021. Contains 348160 sequences. (Running on oeis4.)