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A329275 Expansion of 1 / (1 + Sum_{k>=1} mu(k) * log(1 - 2 * x^k) / k), where mu = A008683.

%I A329275 #8 Nov 11 2019 12:40:41
%S A329275 1,2,5,14,40,116,336,976,2835,8238,23940,69580,202235,587822,1708606,
%T A329275 4966420,14436034,41961830,121972548,354544354,1030574824,2995634338,
%U A329275 8707595956,25310916258,73572844430,213858876100,621637274730,1806952922994,5252386090589,15267448253302
%N A329275 Expansion of 1 / (1 + Sum_{k>=1} mu(k) * log(1 - 2 * x^k) / k), where mu = A008683.
%C A329275 Invert transform of A001037.
%F A329275 a(0) = 1; a(n) = Sum_{k=1..n} A001037(k) * a(n-k).
%p A329275 b:= proc(n) option remember; `if`(n=0, 1, (2^n-add(
%p A329275       d*b(d), d=numtheory[divisors](n) minus {n}))/n)
%p A329275     end:
%p A329275 a:= proc(n) option remember; `if`(n=0, 1,
%p A329275       add(a(n-i)*b(i), i=1..n))
%p A329275     end:
%p A329275 seq(a(n), n=0..30);  # _Alois P. Heinz_, Nov 11 2019
%t A329275 nmax = 29; CoefficientList[Series[1/(1 + Sum[MoebiusMu[k] Log[1 - 2 x^k]/k, {k, 1, nmax}]), {x, 0, nmax}], x]
%t A329275 a[0] = 1; a[n_] := a[n] = Sum[(1/k) DivisorSum[k, MoebiusMu[#] 2^(k/#) &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 29}]
%Y A329275 Cf. A000079, A001037, A008683, A329276.
%K A329275 nonn
%O A329275 0,2
%A A329275 _Ilya Gutkovskiy_, Nov 11 2019