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A216665 Triangular array read by rows: T(n,k) is the number of partitions of n into k parts of 2 different sizes; n>=3, 2<=k<=n-1.

%I A216665 #19 Jun 17 2016 15:23:05
%S A216665 1,1,1,2,2,1,2,1,2,1,3,3,2,2,1,3,3,2,2,2,1,4,3,2,3,2,2,1,4,4,5,1,3,2,
%T A216665 2,1,5,5,3,4,2,3,2,2,1,5,4,4,3,3,2,3,2,2,1,6,6,4,5,2,4,2,3,2,2,1,6,6,
%U A216665 7,5,5,1,4,2,3,2,2,1,7,6,4,3,4,4,2,4,2,3,2,2,1
%N A216665 Triangular array read by rows: T(n,k) is the number of partitions of n into k parts of 2 different sizes; n>=3, 2<=k<=n-1.
%C A216665 Row sums = A002133.
%C A216665 First column (corresponding to k=2) = floor( (n-1)/2 ).
%H A216665 Alois P. Heinz, <a href="/A216665/b216665.txt">Rows n = 3..143, flattened</a>
%H A216665 N. B. Tani and S. Bouroubi, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL14/Tani/tani7.pdf">Enumeration of the Partitions of an Integer into Parts of a Specified Number of Different Sizes and Especially Two Sizes</a>, J. Integer Seqs., Vol. 14 (2011), #11.3.6. (This sequence is Table 1 on p. 10).
%F A216665 G.f.:  Sum_{i>=1} Sum_{j=1..n-1} y^2*x^(i+j)/((1-y*x^j)*(1-y*x^i)).
%e A216665 T(8,3) = 3 because we have: 6+1+1, 4+2+2, 3+3+2.
%e A216665 Triangle indexed from n=3 and k=2:
%e A216665 1;
%e A216665 1, 1;
%e A216665 2, 2, 1;
%e A216665 2, 1, 2, 1;
%e A216665 3, 3, 2, 2, 1;
%e A216665 3, 3, 2, 2, 2, 1;
%e A216665 4, 3, 2, 3, 2, 2, 1;
%e A216665 4, 4, 5, 1, 3, 2, 2, 1;
%t A216665 nn=15;ss=Sum[Sum[y^2 x^(i+j)/(1-y x^i)/(1-y x^j),{j,1,i-1}],{i,1,nn}]; f[list_]:=Select[list,#>0&];Map[f,CoefficientList[Series[ss,{x,0,nn}], {x,y}]]//Flatten
%Y A216665 Cf. A117955, A117956, A274108, A274109.
%K A216665 nonn,tabl
%O A216665 3,4
%A A216665 _Geoffrey Critzer_, Sep 13 2012