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A302550 Expansion of Sum_{k>=1} (-1 + Product_{j>=1} (1 + x^(k*j))^j).

%I A302550 #8 Jun 21 2018 20:02:00
%S A302550 1,3,6,11,17,36,50,94,148,254,386,671,1005,1651,2543,4034,6112,9599,
%T A302550 14410,22178,33189,50196,74485,111591,164149,242967,355317,520817,
%U A302550 755895,1099219,1584520,2285960,3275667,4691845,6682765,9512213,13471240,19059192,26851931,37778822
%N A302550 Expansion of Sum_{k>=1} (-1 + Product_{j>=1} (1 + x^(k*j))^j).
%C A302550 Inverse Moebius transform of A026007.
%H A302550 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A302550 G.f.: Sum_{k>=1} A026007(k)*x^k/(1 - x^k).
%F A302550 a(n) = Sum_{d|n} A026007(d).
%p A302550 with(numtheory):
%p A302550 b:= proc(n) option remember;
%p A302550       add((-1)^(n/d+1)*d^2, d=divisors(n))
%p A302550     end:
%p A302550 g:= proc(n) option remember;
%p A302550       `if`(n=0, 1, add(b(k)*g(n-k), k=1..n)/n)
%p A302550     end:
%p A302550 a:= n-> add(g(d), d=divisors(n)):
%p A302550 seq(a(n), n=1..40);  # _Alois P. Heinz_, Jun 21 2018
%t A302550 nmax = 40; Rest[CoefficientList[Series[Sum[-1 + Product[(1 + x^(k j))^j, {j, 1, nmax}], {k, 1, nmax}], {x, 0, nmax}], x]]
%t A302550 b[n_] := b[n] = SeriesCoefficient[Product[(1 + x^k)^k , {k, 1, n}], {x, 0, n}]; a[n_] := a[n] = SeriesCoefficient[Sum[b[k] x^k/(1 - x^k), {k, 1, n}], {x, 0, n}]; Table[a[n], {n, 40}]
%t A302550 b[0] = 1; b[n_] := b[n] = Sum[Sum[(-1)^(j/d + 1) d^2, {d, Divisors[j]}] b[n - j], {j, n}]/n; a[n_] := a[n] = Sum[b[d], {d, Divisors[n]}]; Table[a[n], {n, 40}]
%Y A302550 Cf. A026007, A047966, A047968, A300276, A302549.
%K A302550 nonn
%O A302550 1,2
%A A302550 _Ilya Gutkovskiy_, Jun 20 2018