%I #13 Jun 01 2012 19:31:32

%S 1,1,1,2,1,1,2,1,1,3,1,1,2,2,2,3,1,1,2,5,1,1,2,2,2,3,2,2,3,5,1,1,1,2,

%T 7,1,1,2,2,2,3,3,3,3,5,3,3,4,4,7,1,1,1,2,4,11,1,1,2,2,2,3,3,3,3,5,4,4,

%U 5,4,7,3,3,3,5,6,11,1,1,1,1,2,4,15

%N Tetrahedron T(j,n,k) in which the slice j is a finite triangle read by rows T(n,k) which lists the sums of the columns of the shell model of partitions with n shells.

%e --------------------------------------------------------

%e Illustration of first five A210952

%e slices of the tetrahedron Row sum

%e --------------------------------------------------------

%e . 1, 1

%e . 1, 1

%e . 1, 2, 3

%e . 1, 1

%e . 1, 2, 3

%e . 1, 1, 3, 5

%e . 1, 1

%e . 1, 2, 3

%e . 2, 2, 3, 7

%e . 1, 1, 2, 5, 9

%e . 1, 1

%e . 1, 2, 3

%e . 2, 2, 3, 7

%e . 2, 2, 3, 5, 12

%e . 1, 1, 1, 2, 7, 12

%e --------------------------------------------------------

%e . 1, 2, 2, 3, 3, 3, 5, 5, 5, 5, 7, 7, 7, 7, 7,

%e Each column sum in the slice j is equal to A000041(j).

%e .

%e Also this sequence can be written as a triangle read by rows in which each row is a flattened triangle. The sequence begins:

%e 1;

%e 1,1,2;

%e 1,1,2,1,1,3;

%e 1,1,2,2,2,3,1,1,2,5;

%e 1,1,2,2,2,3,2,2,3,5,1,1,1,2,7;

%e 1,1,2,2,2,3,3,3,3,5,3,3,4,4,7,1,1,1,2,4,11;

%e 1,1,2,2,2,3,3,3,3,5,4,4,5,4,7,3,3,3,5,6,11,1,1,1,1,2,4,15;

%e Row n has length A000217(n). Row sums give A066186. Right border gives A000041(n), n >= 1.

%Y Cf. A135010, A138121, A209655, A209918, A210952, A210960, A210961.

%K nonn,tabf

%O 1,4

%A _Omar E. Pol_, Apr 24 2012