Re Gray paths cont

[N. J. A. Sloane]

There are two other sequences related to the ones
already mentioned:

%I A003042 M2053
%S A003042 1,2,12,2688,1813091520
%N A003042 Number of Hamiltonian cycles on n-cube.
%D A003042 M. Gardner, Knotted Doughnuts and Other Mathematical Entertainments. Freeman, NY, 1986, p. 24.
%Y A003042 A006069 divided by 2^n.
%K A003042 nonn
%O A003042 1,2
%A A003042 njas

which is clear, and this one, which refers to Gardner's column,
which was the raw materail for the book:

%I A003043 M2112
%S A003043 1,2,18,5712,5859364320
%N A003043 Number of Hamiltonian paths on n-cube.
%D A003043 M. Gardner, Mathematical Games, Sci. Amer. Vol. 228 (No. 4, Apr. 1973), p. 111.
%K A003043 nonn
%O A003043 1,2
%A A003043 njas

Can you see any way to get the latter sequence?

Possibly it is simply wrong, and was silently replaced when he
rewrote the article for the book version?

Neil