A261555 Triangle read by rows: T(n,k) is number of partitions of n having at least k distinct parts (n >= 1, k >= 1).

1, 2, 3, 1, 5, 2, 7, 5, 11, 7, 1, 15, 13, 2, 22, 18, 5, 30, 27, 10, 42, 38, 16, 1, 56, 54, 27, 2, 77, 71, 42, 5, 101, 99, 62, 10, 135, 131, 87, 20, 176, 172, 128, 31, 1, 231, 226, 171, 54, 2, 297, 295, 236, 82, 5, 385, 379, 311, 127, 10, 490, 488, 417, 182, 20

Offset: 1

Author

Michel Marcus, Aug 24 2015

Keywords

• nonn
• tabf

Comments

From Omar E. Pol, Sep 14 2016: (Start)

Row n has length A003056(n) hence the first element of column k is in row A000217(k).

Row sums give A000070.

Alternating row sums give A090794.

Column 1 is A000041, n >= 1. (End)

[0, 0] together with column 2 gives A144300. - Omar E. Pol, Sep 17 2016

Examples

			Triangle starts:
1;
2;
3,  1;
5,  2;
7,  5;
11, 7,  1;
15, 13, 2;
22, 18, 5;
30, 27, 10;
42, 38, 16, 1;
56, 54, 27, 2;
77, 71, 42, 5;
...
		

References

• Jacques Barbot, Essai sur la structuration de l'analyse combinatoire, Paris, Dulac, 1973, Annexe 12A, p. 74.

Links

• Alois P. Heinz, Rows n = 1..500, flattened

Crossrefs

Cf. A000005, A000041, A000070, A000217, A003056, A060177, A090794, A116608, A144300.

Programs

Mathematica

Table[DeleteCases[Map[Count[Map[Length@ Union@ # &, IntegerPartitions@ n], k_ /; k >= #] &, Range@ n], 0], {n, 19}] // Flatten (* Michael De Vlieger, Sep 14 2016 *)

Formula

T(n,k) = Sum_{j>=k} A116608(n,j) assuming A116608(n,j)=0 when j>A003056(n).

T(n,1) - T(n,2) = A000005(n). - Omar E. Pol, Sep 17 2016

Extensions

More terms from Alois P. Heinz, Aug 24 2015