a(n) = {my(walker = 0, nf = n!, res = 0); maxIndex = n; 
while(walker < nf, 
	maxIndex = n; 
	perm = numtoperm(n, walker); 
	if(isperm(perm), 
		res++; 
	); 
	walker += (n - maxIndex)!
); res
}

isperm(v) = {my(tst=vector(#v), qseen = vector(#tst+1)); 
	for(i=1, #v, 
		tst[i]=(i+v[i]*(-1)^v[i])%(#v+1); 
		if(qseen[tst[i] + 1] == 0, 
			qseen[tst[i] + 1] = 1; 
		,
			maxIndex = i; return(0); 
		)
	);
	s=Set(tst);
	if(#s==#v && s[1] == 0 && haswalk(v), 
		return(1)
	);
	0
}

haswalk(v) = {v=concat([0],Vec(v)); my(i=(#v+1)\2+1,t=1); 
while(v[i]!=0,
	i=(i+(v[i]*(-1)^v[i]))%(#v);
	i=(i-1)%#v+1;
	
	t++
);
t==#v
}