A354844 - OEIS

%I #13 Jun 08 2022 15:39:07

%S 1,3,4,29,6,1027,8,26889,272170,861851,12,515592013,14,1530809295,

%T 668366899216,9382044672017,18,1405750464518419,20,

%U 1393382139935385621,4274473667143680022,30537988748467223,24,211745638285336995840025

%N a(n) = n! * Sum_{d|n} (n/d)^d / (d! * (n/d)!).

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

%F If p is prime, a(p) = 1 + p.

%t a[n_] := n! * DivisorSum[n, (n/#)^#/(#! * (n/#)!) &]; Array[a, 25] (* _Amiram Eldar_, Jun 08 2022 *)

%o (PARI) a(n) = n!*sumdiv(n, d, (n/d)^d/(d!*(n/d)!));

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, (exp(k*x^k)-1)/k!)))

%Y Cf. A121860, A354843, A354845.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Jun 08 2022