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A354862 a(n) = n! * Sum_{d|n} (n/d)! / d!.

%I A354862 #25 Aug 30 2023 02:00:25
%S A354862 1,5,37,601,14401,520801,25401601,1626189601,131682257281,
%T A354862 13168407228481,1593350922240001,229442707280223361,
%U A354862 38775788043632640001,7600054676241325858561,1710012252750418295078401,437763137119219420513804801,126513546505547170185216000001
%N A354862 a(n) = n! * Sum_{d|n} (n/d)! / d!.
%F A354862 E.g.f.: Sum_{k>0} k! * (exp(x^k) - 1).
%F A354862 If p is prime, a(p) = 1 + (p!)^2 = A020549(p).
%t A354862 a[n_] := n! * DivisorSum[n, (n/#)! / #! &]; Array[a, 17] (* _Amiram Eldar_, Aug 30 2023 *)
%o A354862 (PARI) a(n) = n!*sumdiv(n, d, (n/d)!/d!);
%o A354862 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, k!*(exp(x^k)-1))))
%o A354862 (Python)
%o A354862 from math import factorial
%o A354862 from sympy import divisors
%o A354862 def A354862(n):
%o A354862     f = factorial(n)
%o A354862     return sum(f*(a := factorial(n//d))//(b:= factorial(d)) + (f*b//a if d**2 < n else 0) for d in divisors(n,generator=True) if d**2 <= n) # _Chai Wah Wu_, Jun 09 2022
%Y A354862 Cf. A020549, A057625, A121860, A354843, A354863.
%K A354862 nonn
%O A354862 1,2
%A A354862 _Seiichi Manyama_, Jun 09 2022