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A144890 Partition number array, called M31hat(5). 3
1, 5, 1, 30, 5, 1, 210, 30, 25, 5, 1, 1680, 210, 150, 30, 25, 5, 1, 15120, 1680, 1050, 900, 210, 150, 125, 30, 25, 5, 1, 151200, 15120, 8400, 6300, 1680, 1050, 900, 750, 210, 150, 125, 30, 25, 5, 1, 1663200, 151200, 75600, 50400, 44100, 15120, 8400, 6300, 5250, 4500 (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) =: M31hat(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,...].

Fourth member (K=5) in the family M31hat(K) of partition number arrays.

If M31hat(5;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle S1hat(5):= A144891.

LINKS

Table of n, a(n) for n=1..54.

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(|S1(5;j,1)|^e(n,k,j),j=1..n) with |S1(5;n,1)| = A049353(n,1) = A001720(n+3) = [1,5,30,210,1680,...] = (n+3)!/4!, 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.

EXAMPLE

[1];[5,1];[30,5,1];[210,30,25,5,1];[1680,210,150,30,25,5,1];...

a(4,3)= 25 = |S1(5;2,1)|^2. The relevant partition of 4 is (2^2).

CROSSREFS

Cf. A144892 (row sums).

Cf. A144885 (M31hat(4) array). A144891 (S1hat(5).

Sequence in context: A188647 A232015 A214882 * A144891 A135892 A049460

Adjacent sequences:  A144887 A144888 A144889 * A144891 A144892 A144893

KEYWORD

nonn,easy,tabf

AUTHOR

Wolfdieter Lang Oct 09 2008, Oct 28 2008

STATUS

approved

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Last modified July 21 02:02 EDT 2019. Contains 325189 sequences. (Running on oeis4.)