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A339884
Triangle read by rows: T(n, m) gives the number of partitions of n with m parts and parts from {1, 2, 3}.
0
1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 2, 1, 1, 0, 0, 2, 2, 2, 1, 1, 0, 0, 1, 3, 2, 2, 1, 1, 0, 0, 1, 2, 3, 2, 2, 1, 1, 0, 0, 0, 2, 3, 3, 2, 2, 1, 1, 0, 0, 0, 1, 3, 3, 3, 2, 2, 1, 1, 0, 0, 0, 1, 2, 4, 3, 3, 2, 2, 1, 1
OFFSET
1,8
COMMENTS
Row sums give A001399(n), for n >= 1.
One could add the column [1,repeat 0] for m = 0 starting with n >= 0.
FORMULA
Sum_{k=0..n} (-1)^k * T(n,k) = A291983(n). - Alois P. Heinz, Feb 01 2021
EXAMPLE
The triangle T(n,m) begins:
n\m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ...
1: 1
2: 1 1
3: 1 1 1
4: 0 2 1 1
5: 0 1 2 1 1
6: 0 1 2 2 1 1
7: 0 0 2 2 2 1 1
8: 0 0 1 3 2 2 1 1
9: 0 0 1 2 3 2 2 1 1
10: 0 0 0 2 3 3 2 2 1 1
11: 0 0 0 1 3 3 3 2 2 1 1
12: 0 0 0 1 2 4 3 3 2 2 1 1
13: 0 0 0 0 2 3 4 3 3 2 2 1 1
14: 0 0 0 0 1 3 4 4 3 3 2 2 1 1
15: 0 0 0 0 1 2 4 4 4 3 3 2 2 1 1
16: 0 0 0 0 0 2 3 5 4 4 3 3 2 2 1 1
17: 0 0 0 0 0 1 3 4 5 4 4 3 3 2 2 1 1
18: 0 0 0 0 0 1 2 4 5 5 4 4 3 3 2 2 1 1
19: 0 0 0 0 0 0 2 3 5 5 5 4 4 3 3 2 2 1 1
20: 0 0 0 0 0 0 1 3 4 6 5 5 4 4 3 3 2 2 1 1
...
Row n = 6: the partitions of 6 with number of parts m = 1,2, ...., 6, and parts from {1,2,3} are (in Abramowitz-Stegun order): [] | [],[],[3,3] | [],[1,2,3],[2^3] | [1^3,3],[1^2,2^2] | [1^4,2] | 1^6, giving 0, 1, 2, 2, 1, 1.
CROSSREFS
Cf. A001399, A008284 (all parts), A145362 (parts 1, 2), A291983.
Compositions: A007818, A030528 (parts 1, 2), A078803 (parts 1, 2, 3).
Sequence in context: A300185 A262804 A048825 * A116375 A301343 A353446
KEYWORD
nonn,tabl,easy
AUTHOR
Wolfdieter Lang, Jan 31 2021
STATUS
approved