|
|
A145975
|
|
Triangle read by rows, partition triangle A027293 convolved with A010815.
|
|
4
|
|
|
1, 1, -1, 2, -1, -1, 3, -2, -1, 0, 5, -3, -2, 0, 0, 7, -5, -3, 0, 0, 1, 11, -7, -5, 0, 0, 1, 0, 15, -11, -7, 0, 0, 2, 0, 1, 22, -15, -11, 0, 0, 3, 0, 1, 0, 30, -22, -15, 0, 0, 5, 0, 2, 0, 0, 42, -30, -22, 0, 0, 7, 0, 3, 0, 0, 0, 56, -42, -30, 0, 0, 11, 0, 5, 0, 0, 0, 0
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,4
|
|
COMMENTS
|
Row sums = [1, 0, 0, 0,...]. (a set of matrix operations equivalent to the comment in A010815 that "convolved with the partition numbers = [1, 0, 0, 0,...].
|
|
LINKS
|
|
|
FORMULA
|
Triangle read by rows, = (A027293 * (A010815 * 0^(n-k)); 0<=k<=n. A027293 = an infinite lower triangular matrix with A000041 in every column (the partition numbers). A010815 = (1, -1, -1, 0, 0, 1,...)
|
|
EXAMPLE
|
First few rows of the triangle =
1;
1, -1;
2, -1, -1;
3, -2, -1, 0;
5, -3, -2, 0, 0;
7, -5, -3, 0, 0, 1;
11, -7, -5, 0, 0, 1, 0;
15, -11, -7, 0, 0, 2, 0, 1;
22, -15, -11, 0, 0, 3, 0, 1, 0;
30, -22, -15, 0, 0, 5, 0, 2, 0, 0;
42, -30, -22, 0, 0, 7, 0, 3, 0, 0, 0;
...
|
|
MATHEMATICA
|
Table[Count[Flatten[Union /@ IntegerPartitions@n], k]*SeriesCoefficient[Product[1 - x^i, {i, k - 1}], {x, 0, k - 1}], {n, 12}, {k, n}] // Flatten (* Robert Price, Jun 15 2020 *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|