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A185370
Triangle read by rows: T(n,k) is the number of occurrences of k in the n-th region of the set of partitions of j, if 1<=n<=A000041(j).
0
1, 1, 1, 2, 0, 1, 0, 1, 3, 1, 0, 1, 0, 0, 1, 5, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 7, 2, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 11, 2, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 2, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 15, 4, 1, 1, 0, 0, 0, 1
OFFSET
1,4
COMMENTS
For the definition of "region of the set of partitions of j" see A206437.
T(n,k) is the number of occurrences of k in the n-th region of the shell model of partitions (see A135010).
T(n,k) is also the number of occurrences of k in the n-th row of triangles A186114, A193870, A206437 (and possibly more).
If the length of row n is a record then the length of row n is j and also A000041(j) = n.
If A000041(j) = n then the sum of the last A187219(j) elements of column k is A182703(j,k) and also the sum of all elements of column k is A066633(j,k).
EXAMPLE
First seven regions of any integer >= 5 are
[1], [2,1], [3,1,1], [2], [4,2,1,1,1], [3], [5,2,1,1,1,1,1] (see illustrations, see also A206437). The 7th region contains five 1's, only one 2 and only one 5. There are no 3's. There are no 4's, so row 7 is [5, 1, 0, 0, 1].
-----------------------------------------
n j m k : 1 2 3 4 5 6 7 8
-----------------------------------------
1 1 1 1;
2 2 1 1, 1;
3 3 1 2, 0, 1;
4 4 1 0, 1;
5 4 2 3, 1, 0, 1;
6 5 1 0, 0, 1;
7 5 2 5, 1, 0, 0, 1;
8 6 1 0, 1;
9 6 2 0, 1, 0, 1;
10 6 3 0, 0, 1;
11 6 4 7, 2, 1, 0, 0, 1;
12 7 1 0, 0, 1;
13 7 2 0, 1, 0, 0, 1;
14 7 3 0, 0, 0, 1;
15 7 4 11, 2, 1, 0, 0, 0, 1;
16 8 1 0, 1;
17 8 2 0, 1, 0, 1;
18 8 3 0, 0, 1;
19 8 4 0, 2, 1, 0, 0, 1;
20 8 5 0, 0, 0, 0, 1;
21 8 6 0, 0, 0, 1;
22 8 7 15, 4, 1, 1, 0, 0, 0, 1;
CROSSREFS
Row n has length A141285(n). Row sums give A194446. Positive terms of column 1 give A000041.
Sequence in context: A238405 A374398 A004173 * A352747 A364955 A112517
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Jan 25 2013
STATUS
approved