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A274109 Triangle read by rows: T(n,k) = number of partitions of n into exactly k parts with exactly two different sizes, the sizes being relatively prime (n >= 3, 2 <= k <= n-1).

%I A274109 #22 Aug 17 2022 10:26:09
%S A274109 1,1,1,2,2,1,1,1,2,1,3,3,2,2,1,2,2,2,2,2,1,3,3,2,3,2,2,1,2,2,4,1,3,2,
%T A274109 2,1,5,5,3,4,2,3,2,2,1,2,2,2,2,3,2,3,2,2,1,6,6,4,5,2,4,2,3,2,2,1,3,3,
%U A274109 5,3,4,1,4,2,3,2,2,1,4,4,3,3,4,4,2,4,2,3,2,2,1,4,4,5,3,4,3,3,2,4,2,3,2,2,1,8,8,5,7,3,5,3,4,2,4,2,3,2,2,1,3,3,5,2,5,2,4,2,4,2,4,2,3,2,2,1,9,9,6,7,3,7,3,4,3,4,2,4,2,3,2,2,1
%N A274109 Triangle read by rows: T(n,k) = number of partitions of n into exactly k parts with exactly two different sizes, the sizes being relatively prime (n >= 3, 2 <= k <= n-1).
%H A274109 N. Benyahia Tani, S. Bouroubi, and O. Kihel, <a href="https://liforce.usthb.dz/sites/default/files/2020-11/article3.pdf">An effective approach for integer partitions using exactly two distinct sizes of parts</a>, Bulletin du Laboratoire 03 (2015), 18-27.
%H A274109 N. Benyahia Tani, S. Bouroubi, and O. Kihel, <a href="http://doi.org/10.4171/EM/326">An effective approach for integer partitions using exactly two distinct sizes of parts</a>, Elemente der Mathematik 72(2) (2017), 66-74.
%e A274109 Triangle T(n,k) (with columns n >= 3 and k >= 2) begins as follows:
%e A274109   1;
%e A274109   1, 1;
%e A274109   2, 2, 1;
%e A274109   1, 1, 2, 1;
%e A274109   3, 3, 2, 2, 1;
%e A274109   2, 2, 2, 2, 2, 1;
%e A274109   3, 3, 2, 3, 2, 2, 1;
%e A274109   2, 2, 4, 1, 3, 2, 2, 1;
%e A274109   5, 5, 3, 4, 2, 3, 2, 2, 1;
%e A274109   2, 2, 2, 2, 3, 2, 3, 2, 2, 1;
%e A274109   6, 6, 4, 5, 2, 4, 2, 3, 2, 2, 1;
%e A274109   3, 3, 5, 3, 4, 1, 4, 2, 3, 2, 2, 1;
%e A274109   4, 4, 3, 3, 4, 4, 2, 4, 2, 3, 2, 2, 1;
%e A274109   ...
%Y A274109 Row sums give A274108.
%Y A274109 Cf. A002133, A117955, A117956, A216665.
%K A274109 nonn,tabl
%O A274109 3,4
%A A274109 _N. J. A. Sloane_, Jun 17 2016