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A067059
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Square array read by antidiagonals of partitions which half fill an n*k box, i.e. partitions of floor[nk/2] or ceiling[nk/2] into up to n positive integers each no more than k.
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12
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1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 3, 3, 1, 1, 1, 1, 3, 5, 5, 3, 1, 1, 1, 1, 4, 6, 8, 6, 4, 1, 1, 1, 1, 4, 8, 12, 12, 8, 4, 1, 1, 1, 1, 5, 10, 18, 20, 18, 10, 5, 1, 1, 1, 1, 5, 13, 24, 32, 32, 24, 13, 5, 1, 1, 1, 1, 6, 15, 33, 49, 58, 49, 33, 15, 6, 1, 1, 1, 1, 6
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,13
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COMMENTS
| The number of partitions of m into up to n positive integers each no more than k is maximised for given n and k by m=floor[nk/2] or ceiling[nk/2] (and possibly some other values).
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EXAMPLE
| Rows start: 1,1,1,1,1,1,...; 1,1,1,1,1,1,...; 1,1,2,2,3,3,...; 1,1,2,3,5,6,...; 1,1,3,5,8,12,...; etc. T(4,5)=12 since 10 can be partitioned into 5+5, 5+4+1, 5+3+2, 5+3+1+1, 5+2+2+1, 4+4+2, 4+3+3, 4+4+1+1, 4+3+2+1, 4+2+2+2, 3+3+3+1, 3+3+2+2.
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MATHEMATICA
| t[n_, k_] := Length[ IntegerPartitions[ Floor[n*k/2], n, Range[k]]]; Flatten[ Table[ t[n-k , k], {n, 0, 13}, {k, 0, n}]] (* From Jean-François Alcover, Jan 02 2012 *)
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CROSSREFS
| As this is symmetric, rows and columns each include A000012 twice, A008619, A001971, A001973, A001975, A001977, A001979 and A001981. Diagonal is A029895. T(n, n*(n-1)) is the magic series A052456.
Sequence in context: A119963 A057790 A052307 * A049704 A047996 A063686
Adjacent sequences: A067056 A067057 A067058 * A067060 A067061 A067062
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KEYWORD
| nonn,tabl
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AUTHOR
| Henry Bottomley (se16(AT)btinternet.com), Feb 17 2002
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