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.

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

Alois P. Heinz, Rows n = 3..143, flattened

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

Cf. A117955, A117956, A274108, A274109.

Sequence in context: A348364 A279620 A278109 * A301384 A258124 A264032

Adjacent sequences: A216662 A216663 A216664 * A216666 A216667 A216668

Keyword

nonn,tabl

Author

Geoffrey Critzer, Sep 13 2012

Status

approved