Maybe Simon was looking at a very old copy of the OEIS.
There are now several exact formulas for A001006.
See the entry for more details.
Here are two:
%F A001006 a(n) = (-1/2) SUM (-3)^i C(1/2,i) C(1/2, j); i+j=n+2, i >= 0, j >= 0.
%F A001006 a(n) = (1/(n+1)) * Sum_{i} (n+1)!/(i!*(i+1)!*(n-2*i)!) - Bernhart.
NJAS