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Search: id:A097305
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| A097305 |
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Array of number of partitions of n with odd parts only and largest part 2*m-1 with m in {1,2,..., ceiling(n/2)}. |
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+0
2
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| 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 2, 1, 1, 1, 3, 3, 2, 1, 1, 3, 4, 2, 1, 1, 1, 4, 4, 3, 2, 1, 1, 4, 5, 4, 2, 1, 1, 1, 4, 6, 5, 3, 2, 1, 1, 5, 7, 6, 4, 2, 1, 1, 1, 5, 8, 7, 5, 3, 2, 1, 1, 5, 9, 9, 6, 4, 2, 1, 1, 1, 6, 10, 10, 8, 5, 3, 2, 1, 1, 6, 11, 12, 10, 6, 4, 2, 1, 1, 1, 6
(list; graph; listen)
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OFFSET
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1,11
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COMMENT
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The sequence of row lengths of this array is A008619 = [1,1,2,2,3,3,4,4,5,5,6,6,7,7,...].
This is the first difference array of A097306.
The number of partitions of N=2*n (n>=1) into even parts with largest part 2*k, with k from 1,..,n, is given by the triangle A008284(n,k).
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LINKS
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W. Lang, First 18 rows.
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FORMULA
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T(n, m)= number of partitions of n with only odd parts and largest part is k:=2*m-1, m=1, 2, ..., ceiling(n/2).
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EXAMPLE
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[1]; [1]; [1,1]; [1,1]; [1,1,1]; [1,2,1]; [1,2,1,1]; [1,2,2,1]; ...
T(8,2)=2 because there are two partitions of 8 with odd parts from {1,3} and 3 appears at least once, namely (1^5,3) and (1^2,3^2).
T(6,2)=2 from 6= 3+3 = 1+1+1+3.
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CROSSREFS
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Row sums: A000009.
Sequence in context: A056731 A042974 A020906 this_sequence A120675 A072699 A143589
Adjacent sequences: A097302 A097303 A097304 this_sequence A097306 A097307 A097308
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KEYWORD
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nonn,tabf,easy
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Aug 13 2004
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