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Factorial equations


Who know how is possible with use Mathematica solving  equations 
involving factorials e.g
Solve[n! + n == 726, n]

My difficult problem to solving is
x=n
y=n!

find function f(x,y) such
f[1,1!]==1, f[2,2!]==2, f[3,3!]==5, f[4,4!]=59,f[5,5!]=246, 
f[6,6!]=1103, f[7,7!]=5247
values are generated by procedure
ClearAll[f]; a = {}; Do[k = 0; Do[k = k + f[n] x^n, {n, 0, s}];
 m = Resultant[k, D[k, x], x]; AppendTo[a, Length[m]], {s, 1, 8}]; a

Who can help in find formula or do above procedure much quickest I will 
be greatfull !
Best wishes
ARTUR



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