A209655 Tetrahedron in which the n-th slice is also one of the three views of the shell model of partitions of A207380 with n shells.

1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 5, 4, 1, 2, 2, 1, 1, 2, 1, 1, 7, 6, 1, 4, 2, 1, 2, 3, 1, 1, 1, 2, 2, 1, 1

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 row 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 last row of each slice gives A008284 and A058398.

It appears that the triangle formed by the first column 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.

Links

Table of n, a(n) for n=1..35.

Example

--------------------------------------------------------
Illustration of first five
slices of the tetrahedron                       Row sum
--------------------------------------------------------
. 1,                                               1
.    2,                                            2
.    1, 1,                                         2
.          3,                                      3
.          2, 1,                                   3
.          1, 1, 1,                                3
.                   5,                             5
.                   4, 1,                          5
.                   2, 2, 1,                       5
.                   1, 2, 1, 1,                    5
.                               7,                 7
.                               6, 1,              7
.                               4, 2, 1,           7
.                               2, 3, 1, 1,        7
.                               1, 2, 2, 1, 1,     7
--------------------------------------------------------
. 1, 3, 1, 6, 2, 1,12, 5, 2, 1,20, 8, 4, 2, 1,
.
Written as a triangle begins:
1;
2, 1, 1;
3, 2, 1, 1, 1, 1;
5, 4, 1, 2, 2, 1, 1, 2, 1, 1;
7, 6, 1, 4, 2, 1, 2, 3, 1, 1, 1, 2, 2, 1, 1;
In which row sums give A066186.

Crossrefs

Column sums give A181187. Main diagonal gives A210765. Another version is A209918.

Cf. A000041, A000217, A002260, A004736, A008284, A026792, A058398, A058399, A066186, A135010, A182703, A182715, A207380.

Sequence in context: A080577 A374515 A302246 * A209918 A030312 A030321

Adjacent sequences: A209652 A209653 A209654 * A209656 A209657 A209658

Keyword

nonn,tabf,more

Author

Omar E. Pol, Mar 25 2012

Status

approved