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

1, 5, 37, 601, 14401, 520801, 25401601, 1626189601, 131682257281, 13168407228481, 1593350922240001, 229442707280223361, 38775788043632640001, 7600054676241325858561, 1710012252750418295078401, 437763137119219420513804801, 126513546505547170185216000001

Offset: 1

Author

Seiichi Manyama, Jun 09 2022

Keywords

• nonn

Crossrefs

Cf. A020549, A057625, A121860, A354843, A354863.

Programs

Mathematica

a[n_] := n! * DivisorSum[n, (n/#)! / #! &]; Array[a, 17] (* Amiram Eldar, Aug 30 2023 *)

PARI

a(n) = n!*sumdiv(n, d, (n/d)!/d!);

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, k!*(exp(x^k)-1))))

Python

from math import factorial
from sympy import divisors
def A354862(n):
    f = factorial(n)
    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

Formula

E.g.f.: Sum_{k>0} k! * (exp(x^k) - 1).

If p is prime, a(p) = 1 + (p!)^2 = A020549(p).