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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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12
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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
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OFFSET
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1,30
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COMMENTS
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The rank of a partition is the largest part minus the number of parts.
The rows are symmetric: for every partition of rank r there is its conjugate with rank -r. [Joerg Arndt, Oct 07 2012]
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REFERENCES
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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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LINKS
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Table of n, a(n) for n=1..105.
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EXAMPLE
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The partition 5 = 4+1 has largest summand 4 and 2 summands, hence has rank 4-2 = 2.
Triangle begins:
[ 1] 1,
[ 2] 1, 0, 1,
[ 3] 1, 0, 1, 0, 1,
[ 4] 1, 0, 1, 1, 1, 0, 1,
[ 5] 1, 0, 1, 1, 1, 1, 1, 0, 1,
[ 6] 1, 0, 1, 1, 2, 1, 2, 1, 1, 0, 1,
[ 7] 1, 0, 1, 1, 2, 1, 3, 1, 2, 1, 1, 0, 1,
[ 8] 1, 0, 1, 1, 2, 2, 3, 2, 3, 2, 2, 1, 1, 0, 1,
[ 9] 1, 0, 1, 1, 2, 2, 3, 3, 4, 3, 3, 2, 2, 1, 1, 0, 1,
[10] 1, 0, 1, 1, 2, 2, 4, 3, 5, 4, 5, 3, 4, 2, 2, 1, 1, 0, 1,
[11] 1, 0, 1, 1, 2, ...
Row 20 is:
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
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Table[ Count[ (First[ # ]-Length[ # ]& /@ Partitions[ k ]), # ]&/(AT)Range[ -k+1, k-1 ], {k, 16} ]
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CROSSREFS
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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
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nonn,nice,tabf,changed
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AUTHOR
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N. J. A. Sloane, Sep 19 2001
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EXTENSIONS
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More terms from Vladeta Jovovic and Wouter Meeussen, Sep 19 2001
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STATUS
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approved
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