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A319130 Expansion of Product_{k>=1} 1/(1 - x^k)^(2^omega(k)), where omega(k) = number of distinct primes dividing k (A001221).

%I A319130 #13 Apr 03 2019 02:54:58
%S A319130 1,1,3,5,10,16,31,47,81,125,203,305,482,710,1082,1582,2348,3380,4933,
%T A319130 7007,10048,14136,19972,27796,38822,53510,73903,101033,138165,187351,
%U A319130 254055,341923,459956,614904,821162,1090740,1447109,1910665,2519325,3308019,4336956
%N A319130 Expansion of Product_{k>=1} 1/(1 - x^k)^(2^omega(k)), where omega(k) = number of distinct primes dividing k (A001221).
%C A319130 Euler transform of A034444.
%H A319130 Vaclav Kotesovec, <a href="/A319130/b319130.txt">Table of n, a(n) for n = 0..10000</a>
%H A319130 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A319130 G.f.: Product_{k>=1} 1/(1 - x^k)^A034444(k).
%F A319130 G.f.: exp(Sum_{k>=1} A048250(k)*x^k/(k*(1 - x^k))).
%F A319130 G.f.: exp(Sum_{k>=1} Sum_{j>=1} mu(j)^2*x^(j*k)/(k*(1 - x^(j*k)))), where mu = Möbius function (A008683).
%F A319130 log(a(n)) ~ sqrt(2*n*log(n)). - _Vaclav Kotesovec_, Sep 13 2018
%p A319130 with(numtheory): a:=series(mul(1/(1-x^k)^(2^nops(factorset(k))),k=1..50),x=0,41): seq(coeff(a,x,n),n=0..40); # _Paolo P. Lava_, Apr 02 2019
%t A319130 nmax = 40; CoefficientList[Series[Product[1/(1 - x^k)^(2^PrimeNu[k]), {k, 1, nmax}], {x, 0, nmax}], x]
%t A319130 nmax = 40; CoefficientList[Series[Exp[Sum[DivisorSum[k, # &, SquareFreeQ[#] &] x^k/(k (1 - x^k)), {k, 1, nmax}]], {x, 0, nmax}], x]
%t A319130 a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d 2^PrimeNu[d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 40}]
%Y A319130 Cf. A001221, A006171, A008683, A034444, A048250, A293548.
%K A319130 nonn
%O A319130 0,3
%A A319130 _Ilya Gutkovskiy_, Sep 11 2018