A288385 Expansion of Product_{k>=1} (1 - x^k)^sigma(k).

1, -1, -3, -1, 0, 10, 8, 12, 1, -28, -29, -67, -51, -28, 79, 163, 256, 343, 273, 136, -351, -649, -1446, -1751, -1889, -1453, -124, 1924, 5138, 7608, 10636, 10903, 10054, 3143, -5799, -20521, -37217, -53057, -65661, -66086, -54430, -15648, 37179, 122732

Offset: 0

Author

Seiichi Manyama, Jun 08 2017

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A001001, A061256.

Product_{k>=1} (1 - x^k)^sigma_m(k): A288098 (m=0), this sequence (m=1), A288389 (m=2), A288392 (m=3).

Programs

Maple

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*sigma(d), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      -add(b(n-i)*a(i), i=0..n-1))
    end:
seq(a(n), n=0..45);  # Alois P. Heinz, Jun 08 2017

Mathematica

b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*DivisorSigma[1, d], {d,
     Divisors[j]}]*b[n - j], {j, 1, n}]/n];
a[n_] := a[n] = If[n == 0, 1, -Sum[b[n - i]*a[i], {i, 0, n - 1}]];
Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Mar 02 2022, after Alois P. Heinz *)

Formula

Convolution inverse of A061256.

a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A001001(k)*a(n-k) for n > 0.

G.f.: exp(-Sum_{k>=1} sigma_2(k)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Oct 29 2018