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%I #20 Oct 31 2021 07:46:29
%S 1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,0,
%T 0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,1,1,1,1,1,1,1,1,1,2,0,0,0,0,0,
%U 0,0,0,0,0,0,0,0,0,0,0,0,0,1
%N Triangle T(n,k,r,u) read by rows: number of partitions of an n X k X r rectangular cuboid on a cubic grid into integer-sided cubes containing u nodes that are unconnected to any of their neighbors, considering only the number of parts; irregular triangle T(n,k,r,u), n >= k >= r >= 1, u >= 0.
%C Row lengths are specified in A228726.
%H Christopher Hunt Gribble, <a href="/A228594/b228594.txt">Rows 1..34 flattened</a>
%H Christopher Hunt Gribble, <a href="/A228594/a228594.cpp.txt">C++ program</a>
%e T(4,4,4,8) = 2 because the 4 X 4 X 4 rectangular cuboid (in this case a cube) has 2 partitions in which there are 8 nodes unconnected to any of their neighbors. The partitions are (8 2 X 2 X 2 cubes) and (37 1 X 1 X 1 cubes and 1 3 X 3 X 3 cube). The partitions and isolated nodes can be illustrated by expanding into 2 dimensions:
%e ._______. ._______. ._______. ._______. ._______.
%e | | | | . | . | | | | | . | . | | | |
%e |___|___| |___|___| |___|___| |___|___| |___|___|
%e | | | | . | . | | | | | . | . | | | |
%e |___|___| |___|___| |___|___| |___|___| |___|___|
%e ._______. ._______. ._______. ._______. ._______.
%e | |_| | . . |_| | . . |_| | |_| |_|_|_|_|
%e | |_| | . . |_| | . . |_| | |_| |_|_|_|_|
%e |_____|_| |_____|_| |_____|_| |_____|_| |_|_|_|_|
%e |_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_|
%e .
%e The irregular triangle begins:
%e u 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ...
%e n k r
%e 1,1,1 1
%e 2,1,1 1
%e 2,2,1 1
%e 2,2,2 1 1
%e 3,1,1 1
%e 3,2,1 1
%e 3,2,2 1 1
%e 3,3,1 1
%e 3,3,2 1 1
%e 3,3,3 1 1 0 0 0 0 0 0 1
%e 4,1,1 1
%e 4,2,1 1
%e 4,2,2 1 1 1
%e 4,3,1 1
%e 4,3,2 1 1 1
%e 4,3,3 1 1 1 0 0 0 0 0 1
%e 4,4,1 1
%e 4,4,2 1 1 1 1 1
%e 4,4,3 1 1 1 1 1 0 0 0 1
%e 4,4,4 1 1 1 1 1 1 1 1 2 0 0 0 0 0 0 0 0 ...
%e 5,1,1 1
%e 5,2,1 1
%e 5,2,2 1 1 1
%e 5,3,1 1
%e 5,3,2 1 1 1
%e 5,3,3 1 1 1 0 0 0 0 0 1 1
%e 5,4,1 1
%e 5,4,2 1 1 1 1 1
%e 5,4,3 1 1 1 1 1 0 0 0 1 1 1
%e 5,4,4 1 1 1 1 1 1 1 1 2 1 1 1 1 0 0 0 0 ...
%e 5,5,1 1
%e 5,5,2 1 1 1 1 1
%e 5,5,3 1 1 1 1 1 0 0 0 1 1 1 1
%e 5,5,4 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 0 0 ...
%Y Row sums = A228202(n,k,r).
%Y Cf. A225542.
%K nonn,tabf
%O 1,63
%A _Christopher Hunt Gribble_, Aug 27 2013
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