an amusing sequence from Belgium!

[Hans Havermann]

> Someone asked me this morning to explain the sequence
> 1, 2, 6, 21, 85, 430, 2586, ...

It's not impossible to make this fit without assuming an error:

f[x_] := f[x] = f[x - 1]*(x - 1) + GCD[3*f[x - 1], (x - 1)]
f[1] = 1;

Table[f[i], {i, 1, 20}]

{1, 2, 6, 21, 85, 430, 2586, 18103, 144825, 1303434, 13034342, 
143377763, 1720533159, 22366931068, 313137034954, 4697055524313, 
75152888389009, 1277599102613154, 22996783847036790, 
436938893093699011}