clearly is clearly

[Wouter Meeussen]

more where that came from:

res=Table[ Length[#]&  /@
      Split[Sort[Apply[Plus,KSubsets[Range[2n], n],1]-n(n+1)/2]], {n,0,9}]
{{1},
 {1,1},
 {1,1,2,1,1},
 {1,1,2,3,3,3,3,2,1,1},
 {1,1,2,3,5,5,7,7,8,7,7,5,5,3,2,1,1},
 {1,1,2,3,5,7,9,11,14,16,18,19,20,20,19,...
 ...

if we could produce the n^2+1 values in the n'th row of "res"
then we could properly continue this sequence:
Length/@(Union /@  res)
{1,1,2,3,6,12,17,24,32,40}
-------------------------------------------------------
we can put Humpty-Dumpty together again:

multi=Table[Range[0,n^2]+n(n+1)/2,{n,0,9}];
Thread[d[res,multi]]/.d->Dot
{0,3,30,210,1260,6930,36036,180180,875160,4157010}

the above is the sum of all the (2n choose n) sums.

        I am sorry, but the terms
        0,3,30,210,1260,6930,36036,180180,875160,4157010 
        do not match anything in the table,
        because it's too simple:
        Table[Binomial[2n,n] n(2n+1)/2,{n,0,9}]
        {0,3,30,210,1260,6930,36036,180180,875160,4157010}


and the average outcome is
{0, 3/2, 5, 21/2, 18, 55/2, 39, 105/2, 68, 171/2}
or   n(2n+1)/2

squeezing the (2n choose n) trials into the n^2+1
different outcome-categories *should* yield something
familiar, and it does.
Dr. Wouter L. J. MEEUSSEN
w.meeussen.vdmcc at vandemoortele.be
eu000949 at pophost.eunet.be