A144269 Partition number array, called M32hat(-1)= 'M32(-1)/M3'= 'A143171/A036040', related to A001497(n-1,m-1)= |S2(-1;n,m)| (generalized Stirling triangle).

1, 1, 1, 3, 1, 1, 15, 3, 1, 1, 1, 105, 15, 3, 3, 1, 1, 1, 945, 105, 15, 9, 15, 3, 1, 3, 1, 1, 1, 10395, 945, 105, 45, 105, 15, 9, 3, 15, 3, 1, 3, 1, 1, 1, 135135, 10395, 945, 315, 225, 945, 105, 45, 15, 9, 105, 15, 9, 3, 1, 15, 3, 1, 3, 1, 1, 1, 2027025, 135135, 10395, 2835

Offset

1,4

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(-1;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(-1;n,k) is summed over those k with fixed number of parts m one obtains triangle S2hat(-1):= A144270(n,m).

Links

Table of n, a(n) for n=1..70.

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(|S2(-1,j,1)|^e(n,k,j),j=1..n) with |S2(-1,n,1)|= A001147(n-1) = (2*n-3)(!^2) (2-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(-1)/M3' = 'A143171/A036040' (elementwise division of arrays).

Example

a(4,3)= 1 = |S2(-1,2,1)|^2. The relevant partition of 4 is (2^2).
[1]; [1,1]; [3,1,1]; [15,3,1,1,1]; [105,15,3,3,1,1,1]; ... [From Wolfdieter Lang, Oct 23 2008]

Crossrefs

Cf. A144271 (M32hat(-2) array).

Sequence in context: A073483 A006956 A072285 * A144270 A110112 A370691

Adjacent sequences: A144266 A144267 A144268 * A144270 A144271 A144272

Keyword

nonn,easy,tabf

Author

Wolfdieter Lang, Oct 09 2008

Extensions

Corrected all entries. Wolfdieter Lang, Oct 23 2008

Status

approved