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a:=series(add(-1+mul(1+j*x^(k*j), j=1..100), k=1..100), x=0, 42): seq(coeff(a, x, n), n=1..41); # Paolo P. Lava, Apr 02 2019

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b[n_] := b[n] = SeriesCoefficient[Product[(1 + k x^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, 41}]

b[0] = 1; b[n_] := b[n] = Sum[Sum[(-d)^(k/d + 1), {d, Divisors[k]}] b[n - k], {k, 1, n}]/n ; a[n_] := a[n] = Sum[b[d], {d, Divisors[n]}]; Table[a[n], {n, 41}]

Cf. A022629, A047966, A300278, A302550, A318025.

allocated for Ilya Gutkovskiy

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

1, 3, 6, 10, 16, 33, 44, 74, 126, 204, 289, 503, 696, 1151, 1749, 2599, 3742, 5928, 8245, 12658, 18351, 26715, 37828, 55296, 78346, 111882, 159664, 226782, 315416, 446670, 618667, 860764, 1199995, 1649820, 2289020, 3157349, 4303996, 5878786, 8033272, 10894516, 14749052

1,2

Inverse Moebius transform of A022629.

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

G.f.: Sum_{k>=1} A022629(k)*x^k/(1 - x^k).

a(n) = Sum_{d|n} A022629(d).

nmax = 41; Rest[CoefficientList[Series[Sum[-1 + Product[(1 + j x^(k j)), {j, 1, nmax}], {k, 1, nmax}], {x, 0, nmax}], x]]

b[n_] := b[n] = SeriesCoefficient[Product[(1 + k x^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, 41}]

b[0] = 1; b[n_] := b[n] = Sum[Sum[(-d)^(k/d + 1), {d, Divisors[k]}] b[n - k], {k, 1, n}]/n ; a[n_] := a[n] = Sum[b[d], {d, Divisors[n]}]; Table[a[n], {n, 41}]

Cf. A022629, A300278, A302550, A318025.

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Ilya Gutkovskiy, Aug 23 2018

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allocated for Ilya Gutkovskiy

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