A145357 Lower triangular array, called S1hat(6), related to partition number array A145356.

1, 6, 1, 42, 6, 1, 336, 78, 6, 1, 3024, 588, 78, 6, 1, 30240, 6804, 804, 78, 6, 1, 332640, 62496, 8316, 804, 78, 6, 1, 3991680, 753984, 85176, 9612, 804, 78, 6, 1, 51891840, 8273664, 1021608, 94248, 9612, 804, 78, 6, 1, 726485760, 109118016, 11394432, 1157688, 102024

Offset

1,2

Comments

If in the partition array M31hat(6):=A145356 entries belonging to partitions with the same parts number m are summed one obtains this triangle of numbers S1hat(6). In the same way the signless Stirling1 triangle |A008275| is obtained from the partition array M_2 = A036039.

The first columns are A001725(n+4), A145359, A145360,...

Links

Table of n, a(n) for n=1..50.

Wolfdieter Lang, First 10 rows of the array and more.

Wolfdieter Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.

Formula

a(n,m) = sum(product(|S1(6;j,1)|^e(n,m,q,j),j=1..n),q=1..p(n,m)) if n>=m>=1, else 0. Here p(n,m)=A008284(n,m), the number of m parts partitions of n and e(n,m,q,j) is the exponent of j in the q-th m part partition of n. |S1(6,n,1)|= A049374(n,1) = A001725(n+4) = (n+4)!/5!.

Example

Triangle begins:
  [1];
  [6,1];
  [42,6,1];
  [336,78,6,1];
  [3024,588,78,6,1];
  ...

Crossrefs

Cf. A145358 (row sums).

Sequence in context: A113365 A293172 A145356 * A035529 A135893 A051338

Adjacent sequences: A145354 A145355 A145356 * A145358 A145359 A145360

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Oct 17 2008

Status

approved