|
|
A162453
|
|
Plane partition triangle, row sums = A000219; derived from the Euler transform of [1, 2, 3, ...].
|
|
1
|
|
|
1, 1, 2, 1, 2, 3, 1, 5, 3, 4, 1, 5, 9, 4, 5, 1, 9, 15, 12, 5, 6, 1, 9, 24, 24, 15, 6, 7, 1, 14, 36, 46, 30, 18, 7, 8, 1, 14, 58, 70, 65, 36, 21, 8, 9, 1, 20, 76, 130, 110, 78, 42, 24, 9, 10, 1, 20, 111, 196, 200, 144, 91, 48, 27, 10, 11, 1, 27, 150, 314, 335, 273, 168, 104, 54, 30
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
COMMENTS
|
Row sums = A000219, number of planar partitions of n starting with offset 1.
|
|
LINKS
|
|
|
FORMULA
|
Construct an array with rows = a, a*b, a*b*c, ...; where a = [1, 1, 1, ...], b = [1, 0, 2, 0, 3, ...], c = [1, 0, 0, 3, 0, 0, 6, ...], d = [1, 0, 0, 0, 4, 0, 0, 0, 10, 0, 0, 0, 20, ...] etc., where rows converge to A000219: (1, 1, 3, 6, 13, 24, ...). The triangle = finite differences of column terms starting from the top.
|
|
EXAMPLE
|
First few rows of the array:
1, 1, 1, 1, 1, 1, ...; = a
1, 1, 3, 3, 6, 6, ...; = a*b
1, 1, 3, 6, 9, 15, ...; = a*b*c
1, 1, 3, 6, 13, 19, ...; = a*b*c*d
1, 1, 3, 6, 13, 24, ...; = a*b*c*d*e
...
then taking finite differences from the top and discarding the first "1" we obtain:
1;
1, 2;
1, 2, 3;
1, 5, 3, 4;
1, 5, 9, 4, 5;
1, 9, 15, 12, 5, 6;
1, 9, 24, 24, 15, 6, 7;
1, 14, 36, 46, 30, 18, 7, 8;
1, 14, 58, 70, 65, 36, 21, 8, 9;
1, 20, 76, 130, 110, 78, 42, 24, 9, 10;
1, 20, 111, 196, 200, 144, 91, 48, 27, 10, 11;
1, 27, 150, 314, 335, 273, 168, 104, 54, 30, 11, 12;
...
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|