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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.
4
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, 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, 7, 5, 5, 1, 4, 2, 3, 2, 2, 1, 7, 6, 4, 3, 4, 4, 2, 4, 2, 3, 2, 2, 1
OFFSET
3,4
COMMENTS
Row sums = A002133.
First column (corresponding to k=2) = floor( (n-1)/2 ).
LINKS
N. B. Tani and S. Bouroubi, Enumeration of the Partitions of an Integer into Parts of a Specified Number of Different Sizes and Especially Two Sizes, J. Integer Seqs., Vol. 14 (2011), #11.3.6. (This sequence is Table 1 on p. 10).
FORMULA
G.f.: Sum_{i>=1} Sum_{j=1..n-1} y^2*x^(i+j)/((1-y*x^j)*(1-y*x^i)).
EXAMPLE
T(8,3) = 3 because we have: 6+1+1, 4+2+2, 3+3+2.
Triangle indexed from n=3 and k=2:
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, 2, 1;
MATHEMATICA
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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Sep 13 2012
STATUS
approved