A144356 Partition number array, called M31(6), related to A049374(n,m)= |S1(6;n,m)| (generalized Stirling triangle).

1, 6, 1, 42, 18, 1, 336, 168, 108, 36, 1, 3024, 1680, 2520, 420, 540, 60, 1, 30240, 18144, 30240, 17640, 5040, 15120, 3240, 840, 1620, 90, 1, 332640, 211680, 381024, 493920, 63504, 211680, 123480, 158760, 11760, 52920, 22680, 1470, 3780, 126, 1, 3991680, 2661120

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) =: M31(6;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,...].

Sixth member (K=6) in the family M31(K) of partition number arrays.

If M31(6;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle |S1(6)|:= A049374.

Links

Table of n, a(n) for n=1..46.

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)=(n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(|S1(6;j,1)|^e(n,k,j),j=1..n)= M3(n,k)*product(|S1(6;j,1)|^e(n,k,j),j=1..n) with |S1(6;n,1)|= A001725(n+4) = (n+4)!/5!, 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. M3(n,k)=A036040.

Example

[1];[6,1];[42,18,1];[336,168,108,36,1];[3024,1680,2520,420,540,60,1];...
a(4,3)= 108 = 3*|S1(6;2,1)|^2. The relevant partition of 4 is (2^2).

Crossrefs

A049402 (row sums).

A144355 (M31(5) array).

Sequence in context: A051338 A062138 A143498 * A049374 A283151 A138192

Adjacent sequences: A144353 A144354 A144355 * A144357 A144358 A144359

Keyword

nonn,easy,tabf

Author

Wolfdieter Lang Oct 09 2008, Oct 28 2008

Status

approved