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

0,2

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

Links

Table of n, a(n) for n=0..73.

Formula

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

Example

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

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 *)

Crossrefs

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

Sequence in context: A134423 A061260 A152097 * A064861 A305299 A308701

Adjacent sequences: A119439 A119440 A119441 * A119443 A119444 A119445

Keyword

nonn,tabl

Author

Alford Arnold, May 19 2006

Extensions

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

Status

approved