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Search: id:A144880
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| A144880 |
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Partition number array, called M31hat(3). |
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+0
4
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| 1, 3, 1, 12, 3, 1, 60, 12, 9, 3, 1, 360, 60, 36, 12, 9, 3, 1, 2520, 360, 180, 144, 60, 36, 27, 12, 9, 3, 1, 20160, 2520, 1080, 720, 360, 180, 144, 108, 60, 36, 27, 12, 9, 3, 1, 181440, 20160, 7560, 4320, 3600, 2520, 1080, 720, 540, 432, 360, 180, 144, 108, 81, 60, 36, 27
(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) =: M31hat(3;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,...].
This is the third (K=3) member of a family of partition number arrays: A107106, A134133,...
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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(|S1(3;j,1)|^e(n,k,j),j=1..n) with |S1(3;n,1)|= A046089(1,n) = [1,3,12,60,...], 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.
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EXAMPLE
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[1];[3,1];[12,3,1];[60,12,9,3,1];[360,60,36,12,9,3,1];...
a(4,3)= 9 = |S1(3;2,1)|^2. The relevant partition of 4 is (2^2).
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CROSSREFS
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A144882 (row sums).
A134133 (M31hat(2) array). A144885 (M31hat(4) array).
Sequence in context: A048522 A118020 A124572 this_sequence A144881 A121420 A117375
Adjacent sequences: A144877 A144878 A144879 this_sequence A144881 A144882 A144883
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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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EXTENSIONS
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In second %e line: changed S1(3;2,1)^2 into |S1(3;2,1)|^2. - Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 17 2008
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