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%I #12 May 23 2012 17:37:15
%S 1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,2,1,1,1,1,1,2,1,1,2,2,1,1,1,2,2,1,
%T 1,1,1,1,2,1,1,3,2,1,1,3,3,2,1,1,1,3,3,2,1,1,1,1,1,2,1,1,3,2,1,1,4,3,
%U 2,1,1,3,4,3,2,1,1,1,3,4,3,2,1,1
%N Tetrahedron T(j,n,k) in which the slice j is a finite triangle read by rows T(n,k) which list the number of parts in the columns of the shell model of partitions with n shells mentioned in A210970.
%e --------------------------------------------------------
%e Illustration of first five
%e slices of the tetrahedron Row sum
%e --------------------------------------------------------
%e . 1, 1
%e . 1, 1
%e . 1, 1, 2
%e . 1, 1
%e . 1, 1, 2
%e . 1, 1, 1, 3
%e . 1, 1
%e . 1, 1, 2
%e . 2, 1, 1, 4
%e . 1, 2, 1, 1, 5
%e . 1, 1
%e . 1, 1, 2
%e . 2, 1, 1, 4
%e . 2, 2, 1, 1, 6
%e . 1, 2, 2, 1, 1, 7
%e --------------------------------------------------------
%e . 1, 2, 1, 3, 2, 1, 5, 4, 2, 1, 7, 6, 4, 2, 1,
%e .
%e It appears that column sums give A058399.
%e Also, written as a triangle read by rows in which each row is a flattened triangle, begins:
%e 1;
%e 1,1,1,
%e 1,1,1,1,1,1;
%e 1,1,1,2,1,1,1,2,1,1;
%e 1,1,1,2,1,1,2,2,1,1,1,2,2,1,1;
%e 1,1,1,2,1,1,3,2,1,1,3,3,2,1,1,1,3,3,2,1,1;
%e 1,1,1,2,1,1,3,2,1,1,4,3,2,1,1,3,4,3,2,1,1,1,3,4,3,2,1,1;
%e In which row sums give A006128.
%Y Cf. A058399, A135010, A138121, A182703, A209655, A209918, A210763, A210961.
%K nonn,tabf
%O 1,14
%A _Omar E. Pol_, Apr 22 2012
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