A119442 Triangle read by rows: row n lists number of unordered partitions of n into k parts which are partition numbers (members of A000041).

1, 2, 1, 3, 2, 1, 5, 7, 2, 1, 7, 11, 7, 2, 1, 11, 26, 19, 7, 2, 1, 15, 40, 38, 19, 7, 2, 1, 22, 83, 78, 54, 19, 7, 2, 1, 30, 120, 168, 102, 54, 19, 7, 2, 1, 42, 223, 301, 244, 134, 54, 19, 7, 2, 1, 56, 320, 557, 471, 292, 134, 54, 19, 7, 2, 1, 77, 566, 1035, 1000, 623, 356, 134, 54

Offset: 0

Author

Alford Arnold, May 19 2006

Keywords

• nonn
• tabl

Comments

A060642 describes the ordered case.

Number of twice-partitions of n of length k. A twice-partition of n is a choice of a partition of each part in a partition of n. - Gus Wiseman, Mar 23 2018

Examples

			Triangle begins:
   1
   2   1
   3   2   1
   5   7   2   1
   7  11   7   2   1
  11  26  19   7   2   1
  15  40  38  19   7   2   1
  22  83  78  54  19   7   2   1
  30 120 168 102  54  19   7   2   1
  42 223 301 244 134  54  19   7   2   1
  56 320 557 471 292 134  54  19   7   2   1
The T(5,3) = 7 twice-partitions: (3)(1)(1), (21)(1)(1), (111)(1)(1), (2)(2)(1), (2)(11)(1), (11)(2)(1), (11)(11)(1). - _Gus Wiseman_, Mar 23 2018
		

Crossrefs

Cf. A000041, A001970, A008284, A036036, A048996, A055887, A061260, A063834, A273873, A281145, A289501, A299200, A299201.

Programs

Mathematica

nn=12;
ser=Product[1/(1-PartitionsP[n]x^n y),{n,nn}];
Table[SeriesCoefficient[ser,{x,0,n},{y,0,k}],{n,nn},{k,n}] (* Gus Wiseman, Mar 23 2018 *)

Formula

G.f.: 1/Product_{k>0} (1-y*A000041(k)*x^k). - Vladeta Jovovic, May 21 2006

Extensions

More terms and better definition from Vladeta Jovovic, May 21 2006