A144274 Partition number array, called M32hat(-2)= 'M32(-2)/M3'= 'A143172/A036040', related to A004747(n,m)= |S2(-2;n,m)| (generalized Stirling triangle).

1, 2, 1, 10, 2, 1, 80, 10, 4, 2, 1, 880, 80, 20, 10, 4, 2, 1, 12320, 880, 160, 100, 80, 20, 8, 10, 4, 2, 1, 209440, 12320, 1760, 800, 880, 160, 100, 40, 80, 20, 8, 10, 4, 2, 1, 4188800, 209440, 24640, 8800, 6400, 12320, 1760, 800, 320, 200, 880, 160, 100, 40, 16, 80, 20

Offset

1,2

Comments

Each partition of n, ordered as in Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to a nonnegative integer a(n,k) =: M32hat(-2;n,k) with the k-th partition of n in A-St order.

The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].

If M32hat(-2;n,k) is summed over those k with fixed number of parts m one obtains triangle S2hat(-2):= A144275(n,m).

Links

Table of n, a(n) for n=1..61.

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,k) = Product_{j=1..n} |S2(-2,j,1)|^e(n,k,j) with |S2(-2,n,1)|= A008544(n-1) = (3*n-4)(!^3) (3-factorials) for n>=2 and 1 if n=1 and the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n.

Formally a(n,k)= 'M32(-2)/M3' = 'A143172/A036040' (elementwise division of arrays).

Example

a(4,3) = 4 = |S2(-2,2,1)|^2. The relevant partition of 4 is (2^2).

Crossrefs

Cf. A144269 (M32hat(-1) array). A144279 (M32hat(-3) array).

Sequence in context: A114692 A112691 A110169 * A144275 A247236 A011268

Adjacent sequences: A144271 A144272 A144273 * A144275 A144276 A144277

Keyword

nonn,easy,tabf

Author

Wolfdieter Lang, Oct 09 2008

Status

approved