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A319111 Expansion of Product_{k>=1} 1/(1 - phi(k)*x^k), where phi = Euler totient function (A000010).

%I A319111 #10 Apr 03 2019 02:55:04
%S A319111 1,1,2,4,7,13,22,38,63,105,174,278,447,707,1122,1766,2729,4213,6482,
%T A319111 9880,15069,22799,34290,51378,76777,114365,169324,250162,368505,
%U A319111 540575,792042,1154798,1680385,2439101,3530308,5103380,7349875,10564955,15155752,21696072,31007949,44199845
%N A319111 Expansion of Product_{k>=1} 1/(1 - phi(k)*x^k), where phi = Euler totient function (A000010).
%H A319111 Vaclav Kotesovec, <a href="/A319111/b319111.txt">Table of n, a(n) for n = 0..8200</a>
%F A319111 G.f.: exp(Sum_{k>=1} Sum_{j>=1} phi(j)^k*x^(j*k)/k).
%F A319111 From _Vaclav Kotesovec_, Feb 08 2019: (Start)
%F A319111 a(n) ~ c * 2^(2*n/5), where
%F A319111 c = 18827.6460615531202942792897255332975807324818737172163... if mod(n,5) = 0
%F A319111 c = 18827.5079339024144115146595255453426552477117955925738... if mod(n,5) = 1
%F A319111 c = 18827.4967567108036710998657106724179082561779712900405... if mod(n,5) = 2
%F A319111 c = 18827.4818413568083742650057347700058389606441225811016... if mod(n,5) = 3
%F A319111 c = 18827.4547665561882994953942505862213438332903500716893... if mod(n,5) = 4
%F A319111 (End)
%p A319111 with(numtheory): a:=series(mul(1/(1-phi(k)*x^k),k=1..50),x=0,42): seq(coeff(a,x,n),n=0..41); # _Paolo P. Lava_, Apr 02 2019
%t A319111 nmax = 41; CoefficientList[Series[Product[1/(1 - EulerPhi[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
%t A319111 nmax = 41; CoefficientList[Series[Exp[Sum[Sum[EulerPhi[j]^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
%t A319111 a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d EulerPhi[d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 41}]
%Y A319111 Cf. A000010, A061255, A159929, A318917.
%Y A319111 Cf. A279784, A306327, A316961.
%K A319111 nonn
%O A319111 0,3
%A A319111 _Ilya Gutkovskiy_, Sep 10 2018