A144353 Partition number array, called M31(3), related to A046089(n,m)= |S1(3;n,m)| (generalized Stirling triangle).

1, 3, 1, 12, 9, 1, 60, 48, 27, 18, 1, 360, 300, 360, 120, 135, 30, 1, 2520, 2160, 2700, 1440, 900, 2160, 405, 240, 405, 45, 1, 20160, 17640, 22680, 25200, 7560, 18900, 10080, 11340, 2100, 7560, 2835, 420, 945, 63, 1, 181440, 161280, 211680, 241920, 126000, 70560, 181440

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(3;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,...].

Third member (K=3) in the family M31(K) of partition number arrays.

If M31(3;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle |S1(3)|:= A046089.

Links

Table of n, a(n) for n=1..51.

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(3;j,1)|^e(n,k,j),j=1..n) = M3(n,k)*product(|S1(3;j,1)|^e(n,k,j),j=1..n) with |S1(3;n,1)|= A001710(n+1) = (n+1)!/2!, 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];[3,1];[12,9,1];[60,48,27,18,1];[360,300,360,120,135,30,1];...
a(4,3)= 27 = 3*|S1(3;2,1)|^2. The relevant partition of 4 is (2^2).

Crossrefs

A049376 (row sums).

A130561 (M31(2) array), A144354 (M31(4) array).

Sequence in context: A243662 A062139 A156366 * A356146 A039811 A046089

Adjacent sequences: A144350 A144351 A144352 * A144354 A144355 A144356

Keyword

nonn,easy,tabf

Author

Wolfdieter Lang Oct 09 2008, Oct 28 2008

Status

approved