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A063995
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Triangle read by rows: T(n,k), n >= 1, -(n-1) <= k <= n-1, = number of partitions of n with rank k.
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9
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1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 3, 1, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2, 2, 3, 2, 3, 2, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2, 2, 3, 3, 4, 3, 3, 2, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2, 2, 4, 3, 5, 4, 5, 3, 4, 2, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,30
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COMMENTS
| The rank of a partition is the largest summand minus the number of summands.
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REFERENCES
| Atkin, A. O. L. and Swinnerton-Dyer, P., Some properties of partitions, Proc. London Math. Soc. (3) 4, (1954). 84-106. Math. Rev. 15,685d.
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EXAMPLE
| The partition 5 = 4+1 has largest summand 4 and 2 summands, hence has rank 4-2 = 2.
1; 1,0,1; 1,0,1,0,1; 1,0,1,1,1,0,1; 1,0,1,1,1,1,1,0,1; ...
T(20, k) = 1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 20, 22, 30, 33, 40, 42, 48, 45, 48, 42, 40, 33, 30, 22, 20, 14, 12, 8, 7, 4, 4, 2, 2, 1, 1, 0, 1; -19 <= k <= 19.
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MATHEMATICA
| Table[ Count[ (First[ # ]-Length[ # ]& /@ Partitions[ k ]), # ]&/(AT)Range[ -k+1, k-1 ], {k, 16} ]
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CROSSREFS
| For the number of partitions of n with rank 0 (balanced partitions) see A047993.
Sequence in context: A052308 A116510 A128915 * A020951 A117118 A117168
Adjacent sequences: A063992 A063993 A063994 * A063996 A063997 A063998
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KEYWORD
| nonn,nice,tabf
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Sep 19 2001
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EXTENSIONS
| More terms from Vladeta Jovovic (vladeta(AT)eunet.rs) and Wouter Meeussen (wouter.meeussen(AT)pandora.be), Sep 19 2001
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