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a(n) = (n-1)! * Sum_{d|n} 1/((d-1)! * (n/d)!^(d-1)).
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%I #10 Feb 23 2024 09:02:34

%S 1,2,3,10,25,156,721,5356,40881,366850,3628801,40048086,479001601,

%T 6228391456,87184121025,1307724593176,20922789888001,355689166978146,

%U 6402373705728001,121645161595446490,2432902128489747201,51090943465394571376,1124000727777607680001

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

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

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

%o (PARI) a(n) = (n-1)!*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, (k-1)!*(exp(x^k/k!)-1))))

%Y Cf. A005225, A087906.

%Y Cf. A370603.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Feb 23 2024