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E.g.f.: Product_{k>=1} 1 / (1 - exp(x) * x^k / k!).
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%I #7 Aug 10 2021 18:48:21

%S 1,1,5,28,205,1856,19964,249005,3535613,56339884,996009280,

%T 19350090365,409850078356,9400728524669,232154433941057,

%U 6141705628777193,173295665869432733,5195039603196754564,164890990869273983108,5524278740902526776085,194815729875439415542760

%N E.g.f.: Product_{k>=1} 1 / (1 - exp(x) * x^k / k!).

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

%F E.g.f.: Product_{k>=1} 1 / (1 - Sum_{j>=k} binomial(j,k) * x^j / j!).

%F a(n) ~ c * n! / ((1 + LambertW(1)) * LambertW(1)^n), where c = Product_{k>=2} (1/(1 - LambertW(1)^(k-1)/k!)) = 1.487589725380080111479849424209442083... - _Vaclav Kotesovec_, Aug 10 2021

%t nmax = 20; CoefficientList[Series[Product[1/(1 - Exp[x] x^k/k!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

%Y Cf. A005651, A140585, A265953, A347006.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Aug 10 2021