A328775 Product_{n>=1} (1 + x^n)^a(n) = 1 + Sum_{n>=1} tau(n) * x^n, where tau = A000005.

1, 2, 0, 2, -1, 1, -1, 2, 1, -2, 0, 2, -1, -2, 2, 3, -2, -1, 2, 1, -4, 0, 3, -1, 3, -3, -2, 0, 1, 9, -15, 3, 17, -13, -1, -1, 9, -2, -18, 27, -10, -14, 24, -17, -15, 24, 27, -43, -37, 72, 43, -116, -11, 147, -98, -24, 67, -56, 24, -44, 213, -258, -193, 707, -435

Offset: 1

Author

Ilya Gutkovskiy, Oct 27 2019

Keywords

• sign

Comments

Inverse weigh transform of A000005.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..2500

Crossrefs

Cf. A000005, A107742, A320779.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= proc(n) option remember; numtheory[tau](n)-b(n, n-1) end:
seq(a(n), n=1..80);  # Alois P. Heinz, Oct 27 2019

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[a[i], j] b[n - i j, i - 1], {j, 0, n/i}]]]; a[n_] := a[n] = DivisorSigma[0, n] - b[n, n - 1]; Array[a, 65]