

A144284


Partition number array, called M32hat(4)= 'M32(4)/M3'= 'A144267/A036040', related to A011801(n,m)= S2(4;n,m) (generalized Stirling triangle).


4



1, 4, 1, 36, 4, 1, 504, 36, 16, 4, 1, 9576, 504, 144, 36, 16, 4, 1, 229824, 9576, 2016, 1296, 504, 144, 64, 36, 16, 4, 1, 6664896, 229824, 38304, 18144, 9576, 2016, 1296, 576, 504, 144, 64, 36, 16, 4, 1, 226606464, 6664896, 919296, 344736, 254016, 229824, 38304, 18144
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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) =: M32hat(4;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,...].
If M32hat(4;n,k) is summed over those k with fixed number of parts m one obtains triangle S2hat(4):= A144285(n,m).


LINKS

Table of n, a(n) for n=1..52.
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(S2(4,j,1)^e(n,k,j),j=1..n) with S2(4,n,1)= A008546(n1) = (5*n6)(!^5) (5factorials) for n>=2 and 1 if 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.
Formally a(n,k)= 'M32(4)/M3' = 'A144267/A036040' (elementwise division of arrays).


EXAMPLE

a(4,3)= 16 = S2(4,2,1)^2. The relevant partition of 4 is (2^2).


CROSSREFS

A144279 (M32hat(3) array). A144341 (M32hat(5) array)
Sequence in context: A051142 A075804 A059844 * A144285 A091741 A061036
Adjacent sequences: A144281 A144282 A144283 * A144285 A144286 A144287


KEYWORD

nonn,easy,tabf


AUTHOR

Wolfdieter Lang Oct 09 2008


STATUS

approved



