login
A173304
Triangle generated from the array in A173302 (partition numbers starting new rows at n = 1, 3, 7, 15, ...).
3
1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 3, 2, 4, 4, 3, 4, 6, 4, 7, 8, 6, 1, 8, 11, 9, 2, 12, 15, 12, 3, 14, 20, 17, 5, 21, 26, 23, 7, 24, 35, 31, 11, 34, 45, 41, 15, 41, 58, 55, 21, 1, 55, 75, 71, 29, 1, 66, 96, 93, 40, 2, 88, 121, 120, 53, 3, 105, 154, 154, 72, 5, 137, 193, 196, 94, 7
OFFSET
0,8
COMMENTS
Row sums = A000041, the partition numbers.
FORMULA
The generating array is in A173302.
1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, ...
1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, ...
1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, ...
1, 1, 2, 3, 5, 7, 11, 15, 22, ...
1, ...
...
Take finite differences from the bottom, creating a new array in which rows are A002865 (a slight variant), A027336, A027338, A027342, ...; i.e., the numbers of partitions of n that do not contain (1, 2, 4, 8, ...) as a part.
EXAMPLE
The finite difference array starts:
1, 1, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 21, 24, ...; = A002865 (a variant)
1, 1, 2, 3, 4, 6, 8, 11, 15, 20, 26, 35, ...; = A027336
1, 1, 2, 3, 4, 6, 9, 12, 17, 23, 31, ...; = A017338
1, 1, 2, 3, 5, 7, 11, ...; = A027342
...
Last, columns of the array become rows of triangle A173304:
1;
1;
1, 1;
2, 2, 1;
2, 3, 2;
4, 4, 3;
4, 6, 4, 1;
7, 8, 6, 1;
8, 11, 9, 2;
12, 15, 12, 3;
14, 20, 17, 5;
21, 26, 23, 7;
24, 35, 31, 11;
34, 45, 41, 15;
41, 58, 55, 21, 1;
55, 75, 71, 29, 1;
66, 96, 93, 40, 2;
88, 121, 120, 53, 3;
105, 154, 154, 72, 5;
137, 193, 196, 94, 7;
...
CROSSREFS
KEYWORD
nonn,tabf,uned
AUTHOR
Gary W. Adamson, Feb 15 2010
STATUS
approved