

A209655


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


5



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
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OFFSET

1,2


COMMENTS

Each slice of the tetrahedron is a triangle, thus the number of elements in the nth slice is A000217(n). The slices are perpendicular to the slices of A026792. Each element of the nth slice equals the volume of a column of the shell model of partitions with n shells. The sum of each row of the nth slice is A000041(n). The sum of all elements of the nth 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 kth parts of all partitions of n in the jth column of rectangle.


LINKS



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;


CROSSREFS

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


KEYWORD

nonn,tabf,more


AUTHOR



STATUS

approved



