A339884 Triangle read by rows: T(n, m) gives the number of partitions of n with m parts and parts from {1, 2, 3}.

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

Author

Wolfdieter Lang, Jan 31 2021

Keywords

• nonn
• tabl
• easy

Comments

Row sums give A001399(n), for n >= 1.

One could add the column [1,repeat 0] for m = 0 starting with n >= 0.

Examples

			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.
		

Links

• Louis Comtet, Advanced Combinatorics, Reidel (1974)

Crossrefs

Cf. A001399, A008284 (all parts), A145362 (parts 1, 2), A232539 (parts <=4), A291983.

Compositions: A007818, A030528 (parts 1, 2), A078803 (parts 1, 2, 3).

Formula

Sum_{k=0..n} (-1)^k * T(n,k) = A291983(n). - Alois P. Heinz, Feb 01 2021

G.f.: 1/((1-u*t)*(1-u*t^2)*(1-u*t^3)). [Comtet page 97 [2c]]. - R. J. Mathar, May 27 2025