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Expansion of Sum_{k>=1} (-1 + Product_{j>=1} (1 + x^(k*j))^j).
2

%I #8 Jun 21 2018 20:02:00

%S 1,3,6,11,17,36,50,94,148,254,386,671,1005,1651,2543,4034,6112,9599,

%T 14410,22178,33189,50196,74485,111591,164149,242967,355317,520817,

%U 755895,1099219,1584520,2285960,3275667,4691845,6682765,9512213,13471240,19059192,26851931,37778822

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

%C Inverse Moebius transform of A026007.

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F G.f.: Sum_{k>=1} A026007(k)*x^k/(1 - x^k).

%F a(n) = Sum_{d|n} A026007(d).

%p with(numtheory):

%p b:= proc(n) option remember;

%p add((-1)^(n/d+1)*d^2, d=divisors(n))

%p end:

%p g:= proc(n) option remember;

%p `if`(n=0, 1, add(b(k)*g(n-k), k=1..n)/n)

%p end:

%p a:= n-> add(g(d), d=divisors(n)):

%p seq(a(n), n=1..40); # _Alois P. Heinz_, Jun 21 2018

%t nmax = 40; Rest[CoefficientList[Series[Sum[-1 + Product[(1 + x^(k j))^j, {j, 1, nmax}], {k, 1, nmax}], {x, 0, nmax}], x]]

%t 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 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 Cf. A026007, A047966, A047968, A300276, A302549.

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, Jun 20 2018