A318290 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

Offset: 1

Author

Ilya Gutkovskiy, Aug 23 2018

Keywords

• nonn

Comments

Inverse Moebius transform of A022629.

Links

• N. J. A. Sloane, Transforms

Crossrefs

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

Programs

Maple

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

Mathematica

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}]

Formula

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

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