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A299069 Expansion of Product_{k>=1} (1 + x^k)^phi(k), where phi() is the Euler totient function (A000010).

%I A299069 #16 Sep 06 2018 17:42:03
%S A299069 1,1,1,3,4,8,11,19,30,44,69,103,157,229,341,491,722,1038,1488,2128,
%T A299069 3015,4267,5989,8407,11713,16289,22523,31097,42729,58569,80003,108957,
%U A299069 147983,200383,270693,364631,490105,656961,878775,1172653,1561626,2074982,2751648,3641536,4810009,6341365,8344967
%N A299069 Expansion of Product_{k>=1} (1 + x^k)^phi(k), where phi() is the Euler totient function (A000010).
%H A299069 Alois P. Heinz, <a href="/A299069/b299069.txt">Table of n, a(n) for n = 0..5000</a>
%H A299069 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A299069 G.f.: Product_{k>=1} (1 + x^k)^A000010(k).
%F A299069 a(n) ~ exp(3^(5/3) * Zeta(3)^(1/3) * n^(2/3) / (2*Pi^(2/3))) * Zeta(3)^(1/6) / (2^(1/3) * 3^(1/6) * Pi^(5/6) * n^(2/3)). - _Vaclav Kotesovec_, Mar 23 2018
%p A299069 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A299069       binomial(numtheory[phi](i), j)*b(n-i*j, i-1), j=0..n/i)))
%p A299069     end:
%p A299069 a:= n-> b(n$2):
%p A299069 seq(a(n), n=0..50);  # _Alois P. Heinz_, Mar 09 2018
%t A299069 nmax = 46; CoefficientList[Series[Product[(1 + x^k)^EulerPhi[k], {k, 1, nmax}], {x, 0, nmax}], x]
%t A299069 a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d EulerPhi[d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 46}]
%Y A299069 Cf. A000010, A061255, A107742, A159929, A192065, A318975.
%K A299069 nonn
%O A299069 0,4
%A A299069 _Ilya Gutkovskiy_, Mar 09 2018