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Search: id:A144284
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| A144284 |
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Partition number array, called M32hat(-4)= 'M32(-4)/M3'= 'A144267/A036040', related to A011801(n,m)= |S2(-4;n,m)| (generalized Stirling triangle). |
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+0
4
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| 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
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1,2
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COMMENT
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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) =: M32hat(-4;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,...].
If M32hat(-4;n,k) is summed over those k with fixed number of parts m one obtains triangle S2hat(-4):= A144285(n,m).
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REFERENCES
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W. Lang, Combinatorial Interpretation of Generalized Stirling Numbers, preprint Oct 2008.
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LINKS
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W. Lang, First 10 rows of the array and more.
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FORMULA
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a(n,k)= product(|S2(-4,j,1)|^e(n,k,j),j=1..n) with |S2(-4,n,1)|= A008546(n-1) = (5*n-6)(!^5) (5-factorials) for n>=2 and 1 if 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.
Formally a(n,k)= 'M32(-4)/M3' = 'A144267/A036040' (elementwise division of arrays).
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EXAMPLE
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a(4,3)= 16 = |S2(-4,2,1)|^2. The relevant partition of 4 is (2^2).
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CROSSREFS
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A144279 (M32hat(-3) array). A144341 (M32hat(-5) array)
Sequence in context: A051142 A075804 A059844 this_sequence A144285 A091741 A061036
Adjacent sequences: A144281 A144282 A144283 this_sequence A144285 A144286 A144287
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KEYWORD
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nonn,easy,tabf
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Oct 09 2008
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