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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}.

%I A339884 #26 May 30 2025 01:09:59
%S A339884 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,
%T A339884 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,
%U A339884 0,1,2,4,3,3,2,2,1,1
%N A339884 Triangle read by rows: T(n, m) gives the number of partitions of n with m parts and parts from {1, 2, 3}.
%C A339884 Row sums give A001399(n), for n >= 1.
%C A339884 One could add the column [1,repeat 0] for m = 0 starting with n >= 0.
%H A339884 Louis Comtet, <a href="https://doi.org/10.1007/978-94-010-2196-8">Advanced Combinatorics</a>, Reidel (1974)
%F A339884 Sum_{k=0..n} (-1)^k * T(n,k) = A291983(n). - _Alois P. Heinz_, Feb 01 2021
%F A339884 G.f.: 1/((1-u*t)*(1-u*t^2)*(1-u*t^3)). [Comtet page 97 [2c]]. - _R. J. Mathar_, May 27 2025
%e A339884 The triangle T(n,m) begins:
%e A339884   n\m  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ...
%e A339884   1:   1
%e A339884   2:   1 1
%e A339884   3:   1 1 1
%e A339884   4:   0 2 1 1
%e A339884   5:   0 1 2 1 1
%e A339884   6:   0 1 2 2 1 1
%e A339884   7:   0 0 2 2 2 1 1
%e A339884   8:   0 0 1 3 2 2 1 1
%e A339884   9:   0 0 1 2 3 2 2 1 1
%e A339884   10:  0 0 0 2 3 3 2 2 1  1
%e A339884   11:  0 0 0 1 3 3 3 2 2  1  1
%e A339884   12:  0 0 0 1 2 4 3 3 2  2  1  1
%e A339884   13:  0 0 0 0 2 3 4 3 3  2  2  1  1
%e A339884   14:  0 0 0 0 1 3 4 4 3  3  2  2  1  1
%e A339884   15:  0 0 0 0 1 2 4 4 4  3  3  2  2  1  1
%e A339884   16:  0 0 0 0 0 2 3 5 4  4  3  3  2  2  1  1
%e A339884   17:  0 0 0 0 0 1 3 4 5  4  4  3  3  2  2  1  1
%e A339884   18:  0 0 0 0 0 1 2 4 5  5  4  4  3  3  2  2  1  1
%e A339884   19:  0 0 0 0 0 0 2 3 5  5  5  4  4  3  3  2  2  1  1
%e A339884   20:  0 0 0 0 0 0 1 3 4  6  5  5  4  4  3  3  2  2  1  1
%e A339884   ...
%e A339884 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.
%Y A339884 Cf. A001399, A008284 (all parts), A145362 (parts 1, 2), A232539 (parts <=4), A291983.
%Y A339884 Compositions: A007818, A030528 (parts 1, 2), A078803 (parts 1, 2, 3).
%K A339884 nonn,tabl,easy
%O A339884 1,8
%A A339884 _Wolfdieter Lang_, Jan 31 2021