%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