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Expansion of e.g.f. Sum_{k>=0} x^k / (k! - k*x^k).
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%I #15 Aug 22 2022 10:06:03

%S 1,1,3,7,37,121,1141,5041,60761,378001,5444461,39916801,729041545,

%T 6227020801,130767460825,1321314894901,31388220966961,355687428096001,

%U 9636906872926477,121645100408832001,3649432697160095561,51223991519836175041,1686001091666419279753

%N Expansion of e.g.f. Sum_{k>=0} x^k / (k! - k*x^k).

%F Expansion of e.g.f. Sum_{k>=0} x^k / (k! * (1 - k*x^k/k!)).

%F a(n) = n! * Sum_{d|n} 1/(d * (d-1)!^(n/d)) for n > 0.

%F a(p) = 1 + p! for prime p.

%t a[n_]:= n! * DivisorSum[n, 1/(# * (# - 1)!^(n/#)) &]; a[0] = 1; Array[a, 23, 0] (* _Amiram Eldar_, Aug 22 2022 *)

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k/(k!-k*x^k))))

%o (PARI) a(n) = if(n==0, 1, n!*sumdiv(n, d, 1/(d*(d-1)!^(n/d))));

%Y Cf. A356029, A356328, A356608.

%Y Cf. A038507, A327578, A356667.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Aug 22 2022