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%I #14 Feb 03 2021 23:21:58
%S 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 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 0,1,2,4,3,3,2,2,1,1
%N Triangle read by rows: T(n, m) gives the number of partitions of n with m parts and parts from {1, 2, 3}.
%C Row sums give A001399(n), for n >= 1.
%C One could add the column [1,repeat 0] for m = 0 starting with n >= 0.
%F Sum_{k=0..n} (-1)^k * T(n,k) = A291983(n). - _Alois P. Heinz_, Feb 01 2021
%e The triangle T(n,m) begins:
%e n\m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ...
%e 1: 1
%e 2: 1 1
%e 3: 1 1 1
%e 4: 0 2 1 1
%e 5: 0 1 2 1 1
%e 6: 0 1 2 2 1 1
%e 7: 0 0 2 2 2 1 1
%e 8: 0 0 1 3 2 2 1 1
%e 9: 0 0 1 2 3 2 2 1 1
%e 10: 0 0 0 2 3 3 2 2 1 1
%e 11: 0 0 0 1 3 3 3 2 2 1 1
%e 12: 0 0 0 1 2 4 3 3 2 2 1 1
%e 13: 0 0 0 0 2 3 4 3 3 2 2 1 1
%e 14: 0 0 0 0 1 3 4 4 3 3 2 2 1 1
%e 15: 0 0 0 0 1 2 4 4 4 3 3 2 2 1 1
%e 16: 0 0 0 0 0 2 3 5 4 4 3 3 2 2 1 1
%e 17: 0 0 0 0 0 1 3 4 5 4 4 3 3 2 2 1 1
%e 18: 0 0 0 0 0 1 2 4 5 5 4 4 3 3 2 2 1 1
%e 19: 0 0 0 0 0 0 2 3 5 5 5 4 4 3 3 2 2 1 1
%e 20: 0 0 0 0 0 0 1 3 4 6 5 5 4 4 3 3 2 2 1 1
%e ...
%e 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 Cf. A001399, A008284 (all parts), A145362 (parts 1, 2), A291983.
%Y Compositions: A007818, A030528 (parts 1, 2), A078803 (parts 1, 2, 3).
%K nonn,tabl,easy
%O 1,8
%A _Wolfdieter Lang_, Jan 31 2021
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