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%I #26 Apr 18 2023 14:45:13
%S 1,1,2,1,3,5,1,4,8,12,1,5,13,24,32,1,6,18,43,73,94,1,7,25,69,141,227,
%T 289,1,8,32,104,252,480,734,910,1,9,41,150,414,920,1656,2430,2934,1,
%U 10,50,207,649,1636,3370,5744,8150,9686,1,11,61,277,967
%N Triangle of numbers a(n,k) = number of balance positions when k equal weights are placed at a k-subset of the points {-n, -(n-1), ..., n-1, n} on a centrally pivoted rod.
%C Also the number of k-subsets of {1..2n-1} with mean n. - _Gus Wiseman_, Apr 16 2023
%D R. E. Odeh and E. J. Cockayne, Balancing weights on the integer line, J. Combin. Theory, 7 (1969), 130-135.
%F Equivalent to number of partitions of n(2k-n+1)/2 into up to n parts each no more than 2k-n+1 so a(n, k)=A067059(n, n(2k-n+1)/2); row sums are A047653(n)-1 = A212352(n). - _Henry Bottomley_, Aug 11 2001
%e From _Gus Wiseman_, Apr 18 2023: (Start)
%e Triangle begins:
%e 1
%e 1 2
%e 1 3 5
%e 1 4 8 12
%e 1 5 13 24 32
%e 1 6 18 43 73 94
%e 1 7 25 69 141 227 289
%e 1 8 32 104 252 480 734 910
%e 1 9 41 150 414 920 1656 2430 2934
%e Row n = 4 counts the following balanced subsets:
%e {0} {-1,1} {-1,0,1} {-3,0,1,2}
%e {-2,2} {-2,0,2} {-4,0,1,3}
%e {-3,3} {-3,0,3} {-2,-1,0,3}
%e {-4,4} {-3,1,2} {-2,-1,1,2}
%e {-4,0,4} {-3,-1,0,4}
%e {-4,1,3} {-3,-1,1,3}
%e {-2,-1,3} {-3,-2,1,4}
%e {-3,-1,4} {-3,-2,2,3}
%e {-4,-1,1,4}
%e {-4,-1,2,3}
%e {-4,-2,2,4}
%e {-4,-3,3,4}
%e (End)
%t a[n_, k_] := Length[ IntegerPartitions[ n*(2k - n + 1)/2, n, Range[2k - n + 1]]]; Flatten[ Table[ a[n, k], {k, 1, 11}, {n, 1, k}]] (* _Jean-François Alcover_, Jan 02 2012 *)
%t Table[Length[Select[Subsets[Range[-n,n]],Length[#]==k&&Total[#]==0&]],{n,8},{k,n}] (* _Gus Wiseman_, Apr 16 2023 *)
%Y Last column is a(n,n) = A002838(n).
%Y Row sums are A212352(n) = A047653(n)-1 = A000980(n)/2-1.
%Y A007318 counts subsets by length, A327481 by mean, A013580 by median.
%Y A327475 counts subsets with integer mean.
%Y Cf. A000975, A024718, A070925, A079309, A326512, A326513, A361801, A362046.
%K nonn,nice,tabl
%O 1,3
%A _N. J. A. Sloane_
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