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Search: id:A144274
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| A144274 |
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Partition number array, called M32hat(-2)= 'M32(-2)/M3'= 'A143172/A036040', related to A004747(n,m)= |S2(-2;n,m)| (generalized Stirling triangle). |
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+0
3
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| 1, 2, 1, 10, 2, 1, 80, 10, 4, 2, 1, 880, 80, 20, 10, 4, 2, 1, 12320, 880, 160, 100, 80, 20, 8, 10, 4, 2, 1, 209440, 12320, 1760, 800, 880, 160, 100, 40, 80, 20, 8, 10, 4, 2, 1, 4188800, 209440, 24640, 8800, 6400, 12320, 1760, 800, 320, 200, 880, 160, 100, 40, 16, 80, 20
(list; graph; listen)
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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(-2;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(-2;n,k) is summed over those k with fixed number of parts m one obtains triangle S2hat(-2):= A144275(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(-2,j,1)|^e(n,k,j),j=1..n) with |S2(-2,n,1)|= A008544(n-1) = (3*n-4)(!^3) (3-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(-2)/M3' = 'A143172/A036040' (elementwise division of arrays).
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EXAMPLE
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a(4,3)= 4 = |S2(-2,2,1)|^2. The relevant partition of 4 is (2^2).
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CROSSREFS
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A144269 (M32hat(-1) array). A144279 (M32hat(-3) array)
Sequence in context: A114692 A112691 A110169 this_sequence A144275 A011268 A163235
Adjacent sequences: A144271 A144272 A144273 this_sequence A144275 A144276 A144277
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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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