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A047997 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.

%I A047997 #30 May 19 2024 08:40:45
%S A047997 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 A047997 289,1,8,32,104,252,480,734,910,1,9,41,150,414,920,1656,2430,2934,1,
%U A047997 10,50,207,649,1636,3370,5744,8150,9686,1,11,61,277,967
%N A047997 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 A047997 Also the number of k-subsets of {1..2n-1} with mean n. - _Gus Wiseman_, Apr 16 2023
%H A047997 R. E. Odeh and E. J. Cockayne, <a href="https://doi.org/10.1016/S0021-9800(69)80047-5">Balancing weights on the integer line</a>, J. Combin. Theory, 7 (1969), 130-135.
%F A047997 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 A047997 From _Gus Wiseman_, Apr 18 2023: (Start)
%e A047997 Triangle begins:
%e A047997     1
%e A047997     1    2
%e A047997     1    3    5
%e A047997     1    4    8   12
%e A047997     1    5   13   24   32
%e A047997     1    6   18   43   73   94
%e A047997     1    7   25   69  141  227  289
%e A047997     1    8   32  104  252  480  734  910
%e A047997     1    9   41  150  414  920 1656 2430 2934
%e A047997 Row n = 4 counts the following balanced subsets:
%e A047997   {0}  {-1,1}  {-1,0,1}   {-3,0,1,2}
%e A047997        {-2,2}  {-2,0,2}   {-4,0,1,3}
%e A047997        {-3,3}  {-3,0,3}   {-2,-1,0,3}
%e A047997        {-4,4}  {-3,1,2}   {-2,-1,1,2}
%e A047997                {-4,0,4}   {-3,-1,0,4}
%e A047997                {-4,1,3}   {-3,-1,1,3}
%e A047997                {-2,-1,3}  {-3,-2,1,4}
%e A047997                {-3,-1,4}  {-3,-2,2,3}
%e A047997                           {-4,-1,1,4}
%e A047997                           {-4,-1,2,3}
%e A047997                           {-4,-2,2,4}
%e A047997                           {-4,-3,3,4}
%e A047997 (End)
%t A047997 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 A047997 Table[Length[Select[Subsets[Range[-n,n]],Length[#]==k&&Total[#]==0&]],{n,8},{k,n}] (* _Gus Wiseman_, Apr 16 2023 *)
%Y A047997 Last column is a(n,n) = A002838(n).
%Y A047997 Row sums are A212352(n) = A047653(n)-1 = A000980(n)/2-1.
%Y A047997 A007318 counts subsets by length, A327481 by mean, A013580 by median.
%Y A047997 A327475 counts subsets with integer mean.
%Y A047997 Cf. A000975, A024718, A070925, A079309, A326512, A326513, A361801, A362046.
%K A047997 nonn,nice,tabl
%O A047997 1,3
%A A047997 _N. J. A. Sloane_