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%I #11 Jan 09 2019 09:17:08
%S 1,1,3,11,53,309,2359,18367,168489,1690217,19416491,233144691,
%T 3187062493,44901291421,700058510943,11509417045799,200586478516049,
%U 3680237286827217,72326917665944659,1467930587827522267,31855597406715020421,718484783876745110021,16993553696264436052103
%N Expansion of e.g.f. Product_{k>=1} 1/(1 - x^k)^(mu(k)^2/k), where mu = Möbius function (A008683).
%F E.g.f.: exp(Sum_{k>=1} 2^omega(k)*x^k/k), where omega(k) = number of distinct primes dividing k (A001221).
%p seq(n!*coeff(series(mul(1/(1-x^k)^(mobius(k)^2/k),k=1..100),x=0,23),x,n),n=0..22); # _Paolo P. Lava_, Jan 09 2019
%t nmax = 22; CoefficientList[Series[Product[1/(1 - x^k)^(MoebiusMu[k]^2/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
%t nmax = 22; CoefficientList[Series[Exp[Sum[2^PrimeNu[k] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
%t a[n_] := a[n] = (n - 1)! Sum[2^PrimeNu[k] a[n - k]/(n - k)!, {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 22}]
%Y Cf. A001221, A008683, A028342, A034444, A073576, A206303, A318913, A318914.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Sep 05 2018
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