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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.
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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