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%I #10 Mar 20 2023 13:08:56
%S 1,1,1,1,2,4,1,3,17,54,1,6,61,892,9235,1,10,220,8159,406653,10538496
%N Triangle read by rows: T(n,k) is the number of ways of dividing an n X k rectangle into integer-sided rectangles, up to rotations and reflections.
%F T(n,k) >= A116694(n,k)/4 if n != k.
%F T(n,n) >= A116694(n,n)/8.
%e Triangle begins:
%e n\k| 0 1 2 3 4 5
%e ---+------------------------------
%e 0 | 1
%e 1 | 1 1
%e 2 | 1 2 4
%e 3 | 1 3 17 54
%e 4 | 1 6 61 892 9235
%e 5 | 1 10 220 8159 406653 10538496
%e The 3 X 2 rectangle can be divided in T(3,2) = 17 inequivalent ways:
%e +---+---+ +---+---+ +---+---+ +---+---+ +---+---+ +---+---+
%e | | | | | | | | | | | | | | | |
%e + + +---+---+ +---+---+ + + + + +---+ + +---+
%e | | | | | | | | | | | | | | |
%e + + + + + + + + + + + + + +---+
%e | | | | | | | | | | | | | | |
%e +---+---+ +---+---+ +---+---+ +---+---+ +---+---+ +---+---+
%e .
%e +---+---+ +---+---+ +---+---+ +---+---+ +---+---+ +---+---+
%e | | | | | | | | | | | | | | | |
%e +---+---+ + + + +---+---+ + +---+ +---+---+ +---+---+
%e | | | | | | | | | | | | | | | |
%e +---+---+ +---+---+ +---+---+ +---+---+ +---+---+ + + +
%e | | | | | | | | | | | | |
%e +---+---+ +---+---+ +---+---+ +---+---+ +---+---+ +---+---+
%e .
%e +---+---+ +---+---+ +---+---+ +---+---+ +---+---+
%e | | | | | | | | | | | | | | |
%e +---+ + +---+---+ +---+---+ +---+---+ +---+---+
%e | | | | | | | | | | | | | |
%e + +---+ +---+---+ + +---+ +---+---+ +---+---+
%e | | | | | | | | | | | | | |
%e +---+---+ +---+---+ +---+---+ +---+---+ +---+---+
%Y Main diagonal: A361524.
%Y Columns: A000012 (k = 0), A005418 (k = 1), A347825 (k = 2; with an exception for n = 2), A361525 (k = 3), A361526 (k = 4).
%Y Cf. A116694 (rotations and reflections are considered distinct), A227690 (square pieces), A360629.
%K nonn,tabl,more
%O 0,5
%A _Pontus von Brömssen_, Mar 15 2023