OFFSET
1,2
COMMENTS
Each slice of the tetrahedron is a triangle, thus the number of elements in the n-th slice is A000217(n). The slices are perpendicular to the slices of A026792. Each element of the n-th slice equals the volume of a column of the shell model of partitions with n shells. The sum of each column of the n-th slice is A000041(n). The sum of all elements of the n-th slice is A066186(n).
It appears that the triangle formed by the first row of each slice gives A058399.
Also consider a vertical rectangle on the infinite square grid with shorter side = n and longer side = p(n) = A000041(n). Each row of rectangle represents a partition of n. Each part of each partition of n is a horizontal rectangle with shorter side = 1 and longer side = k, where k is the size of the part. It appears that T(n,k,j) is also the number of k-th parts of all partitions of n in the j-th column of rectangle.
EXAMPLE
---------------------------------------------------------
Illustration of first five A181187
slices of the tetrahedron Row sum
---------------------------------------------------------
. 1, 1
. 2, 1, 3
. 1, 1
. 3, 2, 1 6
. 1, 1, 2
. 1, 1
. 5, 4, 2, 1, 12
. 1, 2, 2, 5
. 1, 1 2
. 1, 1
. 7, 6, 4, 2, 1, 20
. 1, 2, 3, 2, 8
. 1, 1, 2, 4
. 1, 1, 2
. 1, 1
--------------------------------------------------------
. 1, 2, 2, 3, 3, 3, 5, 5, 5, 5, 7, 7, 7, 7, 7,
.
Note that the 5th slice appears as one of three views of the model in the example section of A207380.
CROSSREFS
KEYWORD
nonn,tabf,more
AUTHOR
Omar E. Pol, Mar 26 2012
STATUS
approved