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

1, 3, 6, 11, 17, 36, 50, 94, 148, 254, 386, 671, 1005, 1651, 2543, 4034, 6112, 9599, 14410, 22178, 33189, 50196, 74485, 111591, 164149, 242967, 355317, 520817, 755895, 1099219, 1584520, 2285960, 3275667, 4691845, 6682765, 9512213, 13471240, 19059192, 26851931, 37778822

Offset: 1

Author

Ilya Gutkovskiy, Jun 20 2018

Keywords

• nonn

Comments

Inverse Moebius transform of A026007.

Links

• N. J. A. Sloane, Transforms

Crossrefs

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

Programs

Maple

with(numtheory):
b:= proc(n) option remember;
      add((-1)^(n/d+1)*d^2, d=divisors(n))
    end:
g:= proc(n) option remember;
      `if`(n=0, 1, add(b(k)*g(n-k), k=1..n)/n)
    end:
a:= n-> add(g(d), d=divisors(n)):
seq(a(n), n=1..40);  # Alois P. Heinz, Jun 21 2018

Mathematica

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

Formula

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

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