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a(n) = n! * Sum_{d|n} 1/((d-1)! * (n/d)^(d-1)).
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%I #9 Feb 23 2024 11:01:47

%S 1,4,9,40,125,1056,5047,51248,383049,4364020,39916811,576885552,

%T 6227020813,99634224704,1334500527375,23592657488416,355687428096017,

%U 7202890599354468,121645100408832019,2679832071577681040,51612375654647808021,1226182612423511392672

%N a(n) = n! * Sum_{d|n} 1/((d-1)! * (n/d)^(d-1)).

%F a(n) = n * A005225(n).

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

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

%o (PARI) a(n) = n!*sumdiv(n, d, 1/((d-1)!*(n/d)^(d-1)));

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

%Y Cf. A370579, A370603.

%Y Cf. A005225.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Feb 23 2024