A144341 Partition number array, called M32hat(-5)= 'M32(-5)/M3'= 'A144268/A036040', related to A011801(n,m)= |S2(-4;n,m)| (generalized Stirling triangle).

1, 5, 1, 55, 5, 1, 935, 55, 25, 5, 1, 21505, 935, 275, 55, 25, 5, 1, 623645, 21505, 4675, 3025, 935, 275, 125, 55, 25, 5, 1, 21827575, 623645, 107525, 51425, 21505, 4675, 3025, 1375, 935, 275, 125, 55, 25, 5, 1, 894930575, 21827575, 3118225, 1182775, 874225, 623645

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(-5;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(-5;n,k) is summed over those k with fixed number of parts m one obtains triangle S2hat(-5):= A144342(n,m).

Links

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

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

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

Formula

a(n,k)= product(|S2(-5,j,1)|^e(n,k,j),j=1..n) with |S2(-5,n,1)|= A008543(n-1) = (6*n-7)(!^6) (6-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(-5)/M3' = 'A144268/A036040' (elementwise division of arrays).

Example

a(4,3)= 25 = |S2(-5,2,1)|^2. The relevant partition of 4 is (2^2).

Crossrefs

A144284 (M32hat(-4) array).

Sequence in context: A049029 A358112 A051150 * A144342 A144268 A013988

Adjacent sequences: A144338 A144339 A144340 * A144342 A144343 A144344

Keyword

nonn,easy,tabf

Author

Wolfdieter Lang Oct 09 2008

Status

approved