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A144353 Partition number array, called M31(3), related to A046089(n,m)= |S1(3;n,m)| (generalized Stirling triangle). 3
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 (list; graph; refs; listen; history; text; internal format)
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 * A039811 A046089 A113360

Adjacent sequences:  A144350 A144351 A144352 * A144354 A144355 A144356

KEYWORD

nonn,easy,tabf

AUTHOR

Wolfdieter Lang Oct 09 2008, Oct 28 2008

STATUS

approved

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Last modified March 22 01:06 EDT 2019. Contains 321406 sequences. (Running on oeis4.)