

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
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OFFSET

1,2


COMMENTS

Each partition of n, ordered as in AbramowitzStegun (ASt order; for the reference see A134278), is mapped to a nonnegative integer a(n,k) =: M31hat(5;n,k) with the kth partition of n in ASt 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 kth partition of n in the ASt 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



