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

%I #6 Nov 11 2019 18:45:39

%S 1,1,2,4,9,20,45,102,232,528,1204,2748,6276,14342,32787,74976,171495,

%T 392337,897696,2054232,4701202,10759689,24627245,56370546,129034271,

%U 295373313,676158166,1547869038,3543458906,8111974160,18570800837,42514665175,97330789942,222825306335

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

%C Invert transform of A000048.

%F a(0) = 1; a(n) = Sum_{k=1..n} A000048(k) * a(n-k).

%t nmax = 33; CoefficientList[Series[1/(1 - Sum[MoebiusMu[2 k] Log[1 - 2 x^k]/(2 k), {k, 1, nmax}]), {x, 0, nmax}], x]

%t a[0] = 1; a[n_] := a[n] = Sum[(1/(2 k)) DivisorSum[k, MoebiusMu[#] 2^(k/#) &, OddQ] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 33}]

%Y Cf. A000048, A008683, A123916, A329275.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Nov 11 2019