[EM] ballots with equalities & truncations allowed

[Warren Smith]

my exponentil generating function 
  exp(x) / (2-exp(x))
gave the counts
1,2,6,26,150,1820
and employing the sloane sequence database found that this sequence also was
previously known ID Number: A000629, however, no connection is mentioned
here to ballots, it arose previously in apparently unconnected contexts.
Interesting.
All kinds of amazing formulas are given for it there.
The constant C is claimed there to be 1/ln(2)=1.4427
as opposed to my earlier estimate of 1.44.  Also the formula
N!/ln(2)^(N+1)   rounded off tot he nearest integer gives the exact answer
for N<=15 and asymptotically but is false when N=16.
Another stunning formula there is
  sum{k>=1}  k^N/2^k .
wds